Constructing quantum spacetime : relation to classical

Perimeter Insitute f. Theoretical Physics

MPI f¨ ur Gravitationsphysik

Institut f¨ ur Physik und Astronomie Universit¨ at Potsdam

Quantum Gravity

Dynamics of Quantum Gravity

31 Caroline St. N.

Am M¨ uhlenberg 1

Karl–Liebknecht Str. 24/25

Waterloo, ON, N2L 2Y5, Kanada

D–14476 Potsdam, Deutschland

D–14476 Potsdam, Deutschland

Constructing quantum spacetime: Relation to classical gravity Publikationsbasierte Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium“ ” (Dr. rer. nat.) in der Wissenschaftsdisziplin: Physik eingereicht an der Mathematisch–Naturwissenschaftlichen Fakult¨at der Universit¨at Potsdam von Sebastian Peter Steinhaus verteidigt am 13. Oktober 2014 vor der Kommission

Prof. Achim Feldmeier Kommissionsvorsitzende Dr. Bianca Dittrich Betreuerin Prof. Martin Wilkens Betreuer Prof. Markus Klein Kommissionsmitglied PD Dr. Carsten Henkel Kommissionsmitglied

Published online at the Institutional Repository of the University of Potsdam: URL http://opus.kobv.de/ubp/volltexte/2015/7255/ URN urn:nbn:de:kobv:517-opus-72558 http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72558

Abstract Despite remarkable progress made in the past century, which has revolutionized our understanding of the universe, there are numerous open questions left in theoretical physics. Particularly important is the fact that the theories describing the fundamental interactions of nature are incompatible. Einstein’s theory of general relative describes gravity as a dynamical spacetime, which is curved by matter and whose curvature determines the motion of matter. On the other hand we have quantum field theory, in form of the standard model of particle physics, where particles interact via the remaining interactions – electromagnetic, weak and strong interaction – on a flat, static spacetime without gravity. A theory of quantum gravity is hoped to cure this incompatibility by heuristically replacing classical spacetime by ‘quantum spacetime’. Several approaches exist attempting to define such a theory with differing underlying premises and ideas, where it is not clear which is to be preferred. Yet a minimal requirement is the compatibility with the classical theory, they attempt to generalize. Interestingly many of these models rely on discrete structures in their definition or postulate discreteness of spacetime to be fundamental. Besides the direct advantages discretisations provide, e.g. permitting numerical simulations, they come with serious caveats requiring thorough investigation: In general discretisations break fundamental diffeomorphism symmetry of gravity and are generically not unique. Both complicates establishing the connection to the classical continuum theory. The main focus of this thesis lies in the investigation of this relation for spin foam models. This is done on different levels of the discretisation / triangulation, ranging from few simplices up to the continuum limit. In the regime of very few simplices we confirm and deepen the connection of spin foam models to discrete gravity. Moreover, we discuss dynamical, e.g. diffeomorphism invariance in the discrete, to fix the ambiguities of the models. In order to satisfy these conditions, the discrete models have to be improved in a renormalisation procedure, which also allows us to study their continuum dynamics. Applied to simplified spin foam models, we uncover a rich, non–trivial fixed point structure, which we summarize in a phase diagram. Inspired by these methods, we propose a method to consistently construct the continuum theory, which comes with a unique vacuum state.

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Abstrakt Trotz bemerkenswerter Fortschritte im vergangenen Jahrhundert, die unser Verst¨andnis des Universums revolutioniert haben, gibt es noch zahlreiche ungekl¨arte Fragen in der theoretischen Physik. Besondere Bedeutung kommt der Tatsache zu, dass die Theorien, welche die fundamentalen Wechselwirkungen der Natur beschreiben, inkompatibel sind. Nach Einsteins allgemeiner Relativit¨atstheorie wird die Gravitation durch eine dynamische Raumzeit dargestellt, die von Materie gekr¨ ummt wird und ihrerseits durch die Kr¨ ummung die Bewegung der Materie bestimmt. Dem gegen¨ uber steht die Quantenfeldtheorie, die die verbliebenen Wechselwirkungen – elektromagnetische, schwache und starke Wechselwirkung – im Standardmodell der Teilchenphysik beschreibt, in dem Teilchen auf einer statischen Raumzeit – ohne Gravitation – miteinander interagieren. Die Hoffnung ist, dass eine Theorie der Quantengravitation diese Inkompatibilit¨at beheben kann, indem, heuristisch, die klassische Raumzeit durch eine Quantenraumzeit‘ ersetzt wird. Es gibt ’ zahlreiche Ans¨ atze eine solche Theorie zu definieren, die auf unterschiedlichen Pr¨amissen und Ideen beruhen, wobei a priori nicht klar ist, welche zu bevorzugen sind. Eine Minimalanforderung an diese Theorien ist Kompatibilit¨ at mit der klassischen Theorie, die sie verallgemeinern sollen. Interessanterweise basieren zahlreiche Modelle in ihrer Definition auf Diskretisierungen oder postulieren eine fundamentale Diskretheit der Raumzeit. Neben den unmittelbaren Vorteilen, die Diskretisierungen bieten, z.B. das Erm¨ oglichen numerischer Simulationen, gibt es auch gravierende Nachteile, die einer ausf¨ uhrlichen Untersuchung bed¨ urfen: Im Allgemeinen brechen Diskretisierungen die fundamentale Diffeomorphismensymmetrie der Gravitation und sind in der Regel nicht eindeutig definiert. Beides erschwert die Wiederherstellung der Verbindung zur klassischen, kontinuierlichen Theorie. Das Hauptaugenmerk dieser Doktorarbeit liegt darin diese Verbindung insbesondere f¨ ur Spin– Schaum–Modelle (spin foam models) zu untersuchen. Dies geschieht auf sehr verschiedenen Ebenen der Diskretisierung / Triangulierung, angefangen bei wenigen Simplizes bis hin zum Kontinuumslimes. Im Regime weniger Simplizes wird die bekannte Verbindung von Spin–Schaum–Modellen zu diskreter Gravitation best¨ atigt und vertieft. Außerdem diskutieren wir dynamische Prinzipien, z.B. Diffeomorphismeninvarianz im Diskreten, um die Ambiguit¨aten der Modelle zu fixieren. Um diese Bedingungen zu erf¨ ullen, m¨ ussen die diskreten Modelle durch Renormierungsverfahren verbessert werden, wodurch wir auch ihre Kontinuumsdynamik untersuchen k¨onnen. Angewandt auf vereinfachte Spin–Schaum–Modelle finden wir eine reichhaltige, nicht–triviale Fixpunkt–Struktur, die wir in einem Phasendiagramm zusammenfassen. Inspiriert von diesen Methoden schlagen wir zu guter Letzt eine konsistente Konstruktionsmethode f¨ ur die Kontinuumstheorie vor, die einen eindeutigen Vakuumszustand definiert.

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Acknowledgements First and foremost, I am indebted to my supervisor Bianca Dittrich. Without her extensive knowledge in quantum gravity (and beyond), her commitment, patience and guidance over almost five years during my time as a diploma and PhD student, I would not be able to present the results of this thesis. Thanks to her I have experienced scientific life in two well–respected research institutes, namely the Albert–Einstein–Institute in Potsdam and the Perimeter Institute f. Theoretical Physics in Waterloo, Canada. I am particularly thankful for the latter, since it also marks my first experience in living and working abroad, from which I have benefited both scientifically and also personally. I am also very grateful for the collaboration with Wojciech Kami´ nski and Mercedes Martin– Benito on our common projects. At this stage, I would also like to thank Benjamin Bahr for offering me a postdoctoral position at Hamburg university, initiating my research career. For their help during my applications, I would also like to acknowledge and thank Bianca Dittrich, Daniele Oriti and Laurent Freidel. During the time I spent at the AEI and PI, I got to know many people, whose company I enjoyed very much. This includes fellow PhD students (and office mates), in particular Johannes Th¨ urigen, Philipp Fleig, Rui Sun, Matti Raasakka, Sylvain Carrozza, Carlos Guedes and Parikshit Dutta at the AEI and Jonathan Ziprick, Jeff Hnybida and Cohl Furey at PI. I would also like to thank Wojciech Kami´ nski, Mercedes Martin–Benito, Daniele Oriti, Laurent Freidel, Lee Smolin, Lorenzo Sindoni, Philipp H¨ ohn, Steffen Gielen, James Ryan, Frank Hellmann, Aristide Baratin, Matteo Smerlak, Valentin Bonzom, Joseph Ben Geloun and Astrid Eichhorn for lots of interesting discussions on (and beyond) physics. I would like to thank the ‘International Max–Planck Research School for Geometric Analysis, Gravitation & String Theory’ and Perimeter Institute for supporting my travels. Also, I would like to acknowledge the German Academic Exchange Service (DAAD) for funding my first year in Canada. At last, I would also like to thank the Perimeter Institute for supporting me during the last year of my PhD studies with an Isaac Newton Chair Graduate Research fellowship. Außerdem m¨ ochte ich mich noch bei meinen guten Freunden in Potsdam, Jens Ole und Katharina, f¨ ur ihre Freundschaft, Unterst¨ utzung und Gastfreundschaft bedanken. Zu guter Letzt gilt mein besonderer Dank meinen Eltern, Gottfried und Marita Steinhaus, f¨ ur ihre bedingungslose und uneingeschr¨ ankte Unterst¨ utzung in unz¨ahligen Situationen, insbesondere in der Zeit meines Umzuges von Deutschland nach Kanada.

Most of my advances were by mistake. You uncover what is when you get rid of what isn’t. – R. Buckminster Fuller

VII

Contents 1 Introduction 1.1 Non–perturbative quantum gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Introduction to Regge calculus . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Spin foam models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Overview of presented manuscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 The Barrett-Crane model: asymptotic measure factor . . . . . . . . . . . . 1.3.3 Path integral measure and triangulation independence in discrete gravity . 1.3.4 Discretization independence implies non–locality in 4D discrete quantum gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Quantum group spin nets: refinement limit and relation to spin foams . . . 1.3.6 Time evolution as refining, coarse graining and entangling . . . . . . . . . .

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2 Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Coherent states and integration kernels . . . . . . . . . . . . . . . . . . . . . 2.1.2 Relation to discrete Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Problem of the next to leading order (NLO) and complete asymptotic expansion 2.1.4 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Modified coherent states, spin-network evaluations and symmetries . . . . . . . . . . 2.2.1 Intertwiners from modified coherent states . . . . . . . . . . . . . . . . . . . . 2.2.2 The action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Symmetry transformations of the action . . . . . . . . . . . . . . . . . . . . . 2.2.4 Groups generated by symmetry transformations . . . . . . . . . . . . . . . . 2.2.5 Geometric lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Normal vectors to the faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.7 Interpretation of planar (spherical) spin-networks as polyhedra . . . . . . . . 2.3 Variable transformation and final form of the integral . . . . . . . . . . . . . . . . . 2.3.1 Partial integration over φ and the new action . . . . . . . . . . . . . . . . . . 2.3.2 Partial integration over φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 New form of the action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Normalization - ‘Theta’ graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Final formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Analysis of 6j symbol and first order Regge Calculus . . . . . . . . . . . . . . . . . . 2.4.1 Stationary point analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Propagator and Hessian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Final Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 First order Regge calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Properties of the next to leading order and complete asymptotic expansion . . . . . 2.5.1 Properties of the Dupuis-Livine form . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Partial integration over φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

2.6 2.A

2.B

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2.E 2.F 2.G 2.H

2.5.3 Different forms of intertwiners and DL property . . . . . . . . . . . 2.5.4 Leading order expansion and a recursion relation for the 6j symbol Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin network evaluation and sign convention . . . . . . . . . . . . . . . . 2.A.1 Penrose prescription for spherical graph . . . . . . . . . . . . . . . 2.A.2 Sign factors and spin structure . . . . . . . . . . . . . . . . . . . . Changes of variables and their Jacobians . . . . . . . . . . . . . . . . . . . 2.B.1 Change of variables ui → ni (integrating out gauge) . . . . . . . . 2.B.2 Variables θ in the flat tetrahedron . . . . . . . . . . . . . . . . . . 2.B.3 Variables θ/ l in the spherical constantly curved tetrahedron . . . 2.B.4 Determinant det ∂θ ∂l for constantly curved simplices . . . . . . . . . Technical computations of determinants . . . . . . . . . . . . . . . . . . . 2.C.1 General knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.C.2 Collection of results . . . . . . . . . . . . . . . . . . . . . . . . . . ∂θ 2.C.3 Computation of det0 ∂lijij . . . . . . . . . . . . . . . . . . . . . . . . Technical computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.D.1 Weak equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.D.2 Proof of the lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.D.3 Theta graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.D.4 Kinetic term in equilateral case . . . . . . . . . . . . . . . . . . . . Dupuis-Livine form and stationary points . . . . . . . . . . . . . . . . . . Stationary point analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation to spin-network kernel formula . . . . . . . . . . . . . . . . . . .

3 The Barrett-Crane model: asymptotic measure factor 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Comments . . . . . . . . . . . . . . . . . . . . . . 3.3 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Stationary point conditions . . . . . . . . . . . . 3.3.2 The hessian . . . . . . . . . . . . . . . . . . . . . 3.4 Technical details . . . . . . . . . . . . . . . . . . . . . . 3.4.1 General facts . . . . . . . . . . . . . . . . . . . . 3.4.2 Limits from curved simplex . . . . . . . . . . . . 3.4.3 Limits of determinant with one null eigenvector . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 3.A Auxiliary definitions and computations . . . . . . . . . . 3.A.1 Definition of the Gram matrix . . . . . . . . . . 3.A.2 Geometric values of lij . . . . . . . . . . . . . . . 3.A.3 Scaling . . . . . . . . . . . . . . . . . . . . . . .

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4 Path integral measure and triangulation independence in 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Linearized Regge Calculus . . . . . . . . . . . . . . . . . . . 4.2.1 Pachner Moves . . . . . . . . . . . . . . . . . . . . . 4.2.2 Pachner moves in 3D . . . . . . . . . . . . . . . . . . 4.2.3 Pachner moves in 4D . . . . . . . . . . . . . . . . . . 4.3 Computation of the Hessian matrix in 3D . . . . . . . . . . 4.3.1 Auxiliary formulas . . . . . . . . . . . . . . . . . . . 4.3.2 Computation of ∂ω ∂l . . . . . . . . . . . . . . . . . . .

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discrete gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents 4.3.3 Summary for the 3–2 move . . . . 4.3.4 Invariance of the path integral . . 4.3.5 4–1 move . . . . . . . . . . . . . . 4.4 Summary for 3D gravity . . . . . . . . . . 4.5 Computation of the Hessian matrix in 4D 4.5.1 4–2 move . . . . . . . . . . . . . . 4.5.2 5–1 move . . . . . . . . . . . . . . 4.5.3 3–3 move . . . . . . . . . . . . . . 4.6 Summary for 4D gravity . . . . . . . . . . 4.7 Discussion . . . . . . . . . . . . . . . . . . 4.A Euclidean integration measure . . . . . . . 4.B Determinant formula for D . . . . . . . .

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5 Discretization independence implies non–locality in 4D ity 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Linearized Regge calculus and previous results . . . . . . 5.3 A geometric interpretation for D . . . . . . . . . . . . . . 5.3.1 Non–locality of the measure . . . . . . . . . . . . . 5.4 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 A D factor absorbing measure for the 5–1 move . . 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.A Special cases of D . . . . . . . . . . . . . . . . . . . . . . 5.A.1 Placing the subdividing vertex far outside . . . . . 5.A.2 Circumcentric subdivision . . . . . . . . . . . . . . 5.B Measure for barycentric subdivision . . . . . . . . . . . . . 5.C Radius of circumscribing sphere . . . . . . . . . . . . . . .

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discrete quantum grav135 . . . . . . . . . . . . . . . 136 . . . . . . . . . . . . . . . 138 . . . . . . . . . . . . . . . 140 . . . . . . . . . . . . . . . 145 . . . . . . . . . . . . . . . 146 . . . . . . . . . . . . . . . 150 . . . . . . . . . . . . . . . 150 . . . . . . . . . . . . . . . 152 . . . . . . . . . . . . . . . 152 . . . . . . . . . . . . . . . 153 . . . . . . . . . . . . . . . 153 . . . . . . . . . . . . . . . 154

6 Quantum group spin nets: refinement limit and relation to spin foams 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Reisenberger’s principle and intertwiner models . . . . . . . . . . . . . . . . 6.3 Quantum group SU(2)k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Basic ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Diagrammatic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Structure of quantum group spin net models . . . . . . . . . . . . . . . . . . 6.4.1 Haar projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Vertex weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The coarse graining algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Symmetry preserving algorithm . . . . . . . . . . . . . . . . . . . . . 6.5.2 Derivation of the flow equation . . . . . . . . . . . . . . . . . . . . . 6.5.3 Geometric interpretation of the flow equation . . . . . . . . . . . . . 6.6 Fixed point intertwiners as initial data and coarse graining results . . . . . 6.6.1 Description of the fixed point intertwiners . . . . . . . . . . . . . . . 6.6.2 Superposition of intertwiners . . . . . . . . . . . . . . . . . . . . . . 6.7 Spin foam interpretation of spin nets and fixed points . . . . . . . . . . . . 6.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.A Additional diagrammatic calculations . . . . . . . . . . . . . . . . . . . . . 6.A.1 Normalization of Haar projector . . . . . . . . . . . . . . . . . . . . 6.A.2 Relation between the two different recoupling schemes . . . . . . . . 6.A.3 Splitting of the 9j-symbol into two 6j-symbols . . . . . . . . . . . .

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159 160 162 167 167 168 170 171 172 173 175 176 178 180 180 186 189 191 193 193 194 194

XI

Contents 7 Time evolution as refining, coarse graining and entangling 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Time evolving phase spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Example: evolution of a scalar field on an extending triangulation . . . . . 7.2.2 Refinement as adding degrees of freedom in the vacuum state . . . . . . . . 7.2.3 Massless scalar field in a 2D Lorentzian space time . . . . . . . . . . . . . . 7.2.4 Refinement moves in simplicial gravity: adding gauge and vacuum degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Refining in quantum theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Inductive limit construction of a continuum Hilbert space . . . . . . . . . . 7.3.2 Refining time evolution and no–boundary vacuum . . . . . . . . . . . . . . 7.3.3 Path independence of evolution and consistent embedding maps . . . . . . 7.3.4 Pre–constraints and coarse graining . . . . . . . . . . . . . . . . . . . . . . 7.4 How to define a continuum theory of quantum gravity . . . . . . . . . . . . . . . . 7.5 Tensor network coarse graining: time evolution in radial direction . . . . . . . . . . 7.5.1 Radial evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Truncations via singular value decomposition . . . . . . . . . . . . . . . . . 7.5.3 Embedding maps and truncations . . . . . . . . . . . . . . . . . . . . . . . 7.5.4 Embedding maps for the fixed points . . . . . . . . . . . . . . . . . . . . . . 7.6 Topological theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Time evolution as coarse graining and refining . . . . . . . . . . . . . . . . 7.6.2 Consistent embedding maps and inductive limit . . . . . . . . . . . . . . . . 7.6.3 3D topological theories and entangling moves . . . . . . . . . . . . . . . . . 7.6.4 Constructing inductive limit Hilbert spaces for non–topological theories . . 7.7 Geometric interpretation of the refining maps . . . . . . . . . . . . . . . . . . . . . 7.7.1 2D example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.2 Spin foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

203 204 205 207 210 212 213

. . . . . . . . . . . . . . . . . . . . .

215 216 217 217 220 221 222 224 225 226 226 227 228 229 230 231 231 232 232 233 234

8 Discussion 8.1 Microscopic building blocks: asymptotic expansion and the measure factor . . 8.1.1 Modified coherent states and the relation to first order Regge calculus 8.1.2 Measure factor for a 4D spin foam model . . . . . . . . . . . . . . . . 8.1.3 What is the interpretation of the asymptotic expansion? . . . . . . . . 8.2 Lectures from triangulation independence: Regge calculus . . . . . . . . . . . 8.2.1 Triangulation independence implies non–locality . . . . . . . . . . . . 8.3 A change of perspective: Tensor network renormalization . . . . . . . . . . . 8.3.1 Coarse graining spin nets: finite groups . . . . . . . . . . . . . . . . . 8.3.2 Quantum group spin nets . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Embedding maps as time evolution . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Cylindrical consistency and the physical vacuum . . . . . . . . . . . . 8.4.2 Constructing quantum space time . . . . . . . . . . . . . . . . . . . . 8.4.3 How to (possibly) get propagating degrees of freedom . . . . . . . . .

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247 247 248 249 250 251 253 254 257 258 260 261 263 265

9 Summary and conclusion

XII

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267

1 Introduction Many technological advances achieved in the last one hundred years, which have greatly changed and influenced human life, go back to the discovery of two fundamental physical theories: on the one hand, we have quantum theory, i.e. quantum mechanics and quantum field theory, describing nature on small scales, e.g. subatomic particles interacting via the electromagnetic, strong and weak forces encoded in the so–called Standard Model. Quantum theories not only solved long lasting problems, e.g. the ultraviolet catastrophe of electromagnetism, and explained the stability of atoms, they stimulated condensed matter physics, which lead to the development of computers and with it our modern communication society with all of its opportunities and perils. Additionally, it gave rise to classically unknown effects such as entanglement, which opened up completely new fields of research and may spark another revolution of its own, namely quantum computing. On the other hand, we have general relativity, the classical theory of gravity, the last and comparably weakest of the known fundamental interactions in nature. Thus it is mostly associated to interactions at very large scales to describe the dynamics of celestial bodies. General relativity, together with special relativity, revolutionized our understanding of space and time: In contrast to Newton’s theory of gravity, it does not consider an absolute space and time as a given background structure on which gravitational interactions take place. Instead space and time are tied together into a dynamical spacetime, whose curvature, induced by matter, describes then again the universal gravitational interaction between massive objects. In short, the gravitational interactions are encoded into the dynamical geometry of spacetime. Hence, there exists no preferential choice of spacetime or coordinate system, which gives rise to the fundamental symmetry of general relativity, namely diffeomorphism symmetry, also known as general covariance. In brief, the dynamics of gravity are invariant under all smooth bijective maps from the underlying manifold to itself, such that only diffeomorphism invariant quantities are physically relevant. Of course general relativity can claim its personal share in novel physical predictions, to name a few such as the perihelion precession of Mercury and bending of light rays by heavy objects in the sky, known as gravitational lensing. From a technological perspective, it contributed the necessary corrections to Newtonian gravity for the Global Positioning System (GPS) to work flawlessly. An inquiring phenomenon discovered as a particular solution to Einstein’s equations are black holes, special regions in spacetime characterized by mass, angular momentum, etc. that causally disconnect their interior from the surrounding spacetime by an event horizon. These objects, whose existence still has to be confirmed directly, e.g. by the direct detection of gravitational waves, are a topic of active research. Remarkably, both theories are very well tested and still mark the most predictive theories developed in theoretical physics. To mention a very recent result at the LHC, the highly celebrated detection of (probably) the Higgs boson experimentally [1, 2] confirms the last missing piece in the Standard Model of particle physics, whereas, up to now, no indication of supersymmetry, extra dimensions or a fifth force have been observed. On the gravitational side, experiments rather appear to put more and more strict bounds onto alternative theories of gravity or (conjectured) quantum gravity effects, yet new experiments might uncover novel effects challenging general relativity, see [3] for a recent review. Very recent results from cosmological experiments, such as the BICEP2 experiment [4], which detected an imprint of primordial gravitational waves in the cosmic microwave background, appear promising to blaze the trail towards new insights in cosmology, which will hopefully affect quantum gravity. Indeed, quantum field theory and classical gravity are physically well established theories that describe the phenomena currently experimentally accessible to us very well. However, this neither

1

1 Introduction implies that no new physical effects will be uncovered at higher energy scales nor does it mean that our understanding of the theories are complete, e.g this can concern the values of the parameters of the theories, their interpretation and origin: the cosmological constant is a prime example. Originally introduced by Einstein in 1917 to model a static universe, it has been measured to be positive, yet very small, and it is a vital parameter in modern models of cosmology [5, 6]. However we are lacking a convincing explanation of its origin and value; an attempt well–known for its obvious failure is the explanation of the cosmological constant as the zero–point energy of quantum fields, cut–off at the Planck scale, which gives a result roughly 120 orders of magnitude too large. This is usually referred to as the ‘Cosmological constant problem’ [7]. In fact, it is frequently argued that many of the current shortcomings in our understanding of nature is rooted in the fact that we do not have a theory of quantum gravity. Indeed, such a theory is necessary for two reasons: First, general relativity is not a complete theory, since it exhibits singularities, e.g. at the origin of our universe or other cosmological scenarios like black holes, where either curvature or energy density diverges and the theory loses its validity. Generically, this is expected to happen at energies close to the Planck scale. Second, and even more troubling, classical gravity is incompatible with quantum theories: indeed, the two most tested and predictive theories are very well understood on scales, where either one of the two is negligible. On scales of the size of a nucleus, gravity is insignificantly weak in comparison to strong, weak and electromagnetic force, whereas on scales of the solar system (and beyond), the weak and strong force are non–relevant and the electromagnetic interactions are well approximated by the classical theory (or semi–classical approximations). However in cosmological situations like the Big Bang, the cosmological singularity at the beginning of the universe, matter and gravitational interactions (and backreactions) are crucial and require compatible theories. The incompatibility between general relativity and quantum field theory is caused by opposing initial assumptions and different technical ingredients: In general relativity, spacetime itself, i.e. the metric on a (differential) manifold, is dynamical and determined by the Einstein equations of motion, in which matter is universally coupled to gravity. As a result, it is not reasonable to distinguish one particular spacetime as a special background, nor should physics depend on the special choice of spacetime coordinates. This is encoded as diffeomorphism invariance in GR, implemented as a symmetry of the Einstein–Hilbert action, which is deeply intertwined with the dynamics of gravity: e.g. in the canonical formalism, gravity is totally constrained, i.e. the Hamiltonian governing time evolution is a constraint itself and forced to vanish. That means that time evolution, generated by the constraint, itself is a gauge transformation, which is often referred to as the ‘problem of time’ or ‘frozen time formalism’ [8–11]. Hence, diffeomorphism symmetry is an essential ingredient of the classical theory of gravity and can serve as a test or guiding principle in the construction of a quantum theory of gravity. Indeed, as we will see the fate of diffeomorphism symmetry plays an important role in the definition of non–perturbative quantum gravity theories. In contrast to general relativity, quantum field theory is defined on one particular spacetime, which can be seen as the ‘stage’ on which the dynamics of quantum fields takes place. This background spacetime, frequently chosen to be flat Minkowski space, is fixed and not dynamical, i.e. spacetime is not interacting with the matter content. In other words, (full) gravity is excluded from the dynamics by construction. In many scenarios, like collider experiments on earth, this is a very good approximation, since the involved energies and particle masses are too small to get non–negligible backreaction effects. Gravity can be partially, or rather passively, incorporated by choosing a curved background spacetime, an essential generalization to derive Hawking radiation of black holes1 . Indeed fixing a background spacetime is essential in providing the technical tools for 1

2

Since a black hole emits thermal radiation, it will eventually evaporate due to energy conservation. However, this process is far from understood and a very active, and controversially discussed, area of research. In particular the information paradox is investigated: Information that is hidden behind the event horizon of the black hole, e.g. matter that has fallen in, gives rise to an entanglement entropy of the black hole. Under evaporation, this information should be released, however, if only thermal radiation is emitted, how can this thermal radiation

1.1 Non–perturbative quantum gravity the (perturbative) (Fock–)quantization of fields and computing scattering processes of excitations of the quantum fields. Hence, the most straightforward idea to define a quantum field theory of gravity and matter is to drop background independence of gravity: Instead of quantizing the full gravitational field, i.e. the metric, one quantizes the (weak) fluctuations around a fixed background, e.g. Minkowski space, as a typical quantum field theory in a perturbative expansion in Feynman diagrams. Classically this approach is known as linearized gravity and describes gravitational waves. The quantum excitations of gravitational waves are particles of spin 2, called gravitons, that interact universally with all other matter fields and, more importantly, with themselves. However this theory, also for pure gravity, is not predictive: As for any quantum field theory, there exist Feynman diagrams that diverge, however these divergences can be absorbed by renormalizing the theory and performing a measurement for each independent parameter of the theory. Yet in perturbative quantum gravity, the divergences cannot be absorbed into a finite number of independent couplings, e.g. the first non–renormalizable term has been identified at the two–loop level [12, 13]. This can either be inferred from the dimensionality of the gravitational coupling constant, which comes with a positive mass dimension. As a result, this form of perturbative quantum gravity is called perturbatively non–renormalizable. There exist several research fields that explore possible non–perturbative theories of quantum gravity to circumvent or solve this issue. Roughly these can be split into two categories: the first relies on the idea that gravity cannot be consistently quantized just by itself, but should rather be contained in a unified theory with all other interactions. String theory typically falls into this category, which however relies on a background spacetime (or sometimes a background boundary spacetime in the AdS / CFT correspondence). The second one, which we will focus on in this thesis, deals with the non–perturbative quantization of pure gravity, characterized by preserving the dynamical nature of spacetime, to which the matter content and interactions are coupled. Tentatively, one might be tempted to call this approach ‘constructing quantum spacetime’.

1.1 Non–perturbative quantum gravity The term ‘non–perturbative quantum gravity’ joins several different approaches, which can be well distinguished by considering their main underlying ideas. Asymptotic safety [14, 15] assumes that gravity can be formulated as a quantum field theory, since it is expected that the renormalization group flow, i.e. the change of the fundamental coupling constants of the theory under integration of higher momentum–modes, has a non–trivial (non–Gaussian) fixed point in the ultraviolet, which is also called the UV completion of the theory. Or take causal dynamical triangulations [16], a numerical approach simulating quantum spacetime by summing over all possible (equilateral) triangulations weighted by the Regge action [17, 18], a form of discrete gravity, which exhibits a phase (in parameter space) in which the three–volume of spatial slices (on average) resembles a de Sitter spacetime. One of the best–known approaches is loop quantum gravity [19, 20], a background independent attempt to canonically quantize gravity, in which the spin–network states, i.e. states in the kinematical Hilbert space, live on discrete graphs embedded in a 3D hypersurface. Physical information is contained in the physical states, i.e. all states that are annihilated by the quantum constraint operators. The covariant approach related to loop quantum gravity is known as spin foam models [21, 22], which are defined on a 2–complex and can be understood as discrete versions of the gravitational path integral. E.g. they give the transition amplitudes between two 3–geometries by assigning amplitudes to the connecting discrete 4–geometry, where it has been shown that the amplitudes assigned to (one of the) basic building blocks are proportional to the discrete Regge action in a semi–classical limit [23–29], see also the papers [30, 31] presented in chapters 2 and 3. contain this information? What is the process that reduces the entanglement entropy under evaporation?

3

1 Introduction A related approach is group field theory (GFT) [32, 33], which defines a field theory on a group, generating pre–geometric complexes via its perturbative expansion in Feynman diagrams. Group field theories are related to spin foam models, since their Feynman diagrams actually resemble spin foam amplitudes. This little excursion is meant to illustrate the variety of different approaches that can be summarized under the topic ‘quantum gravity’. Interestingly, several approaches listed above have common ideas, at least at the conceptual level: As quantum field theories, asymptotic safety and group field theory explicitly examine the renormalization group flow of the theory, namely whether the theory is renormalizable, i.e. there exists a scheme to absorb the divergences of the theory, and how the dynamics of the theory changes at different energy scales. On the other hand, loop quantum gravity2 , spin foam models, causal dynamical triangulations and group field theories rely on discrete structures in their construction, which serve as a truncation of the number of degrees of freedom and allow to non–perturbatively define the dynamics. In some approaches to quantum gravity the discreteness of spacetime is postulated to be fundamental, take for example causal set theory [36]. One goal of this thesis is to define and develop a renormalization group method for discrete approaches, to be more concrete for (analogue) spin foam models, so–called spin nets [37–39], in order to examine the (effective) dynamics of the theory depending on the number of discrete building blocks involved. The hopes are on the one hand that this scheme helps to fix the ambiguities (and pathologies) involved in the definition of the discrete theory, while on the other hand it allows us to explore the different phases of the theory and, furthermore, give a consistent construction method for the associated continuum theory. This goal is very ambitious, but a crucial point if we consider discretisations, in particular in gravitational theories. Despite their desirable properties, the introduction of discretisations (in both classical and quantum theories) has serious repercussions: • Discretisations of a continuous action are usually not uniquely defined, that means one can define a plethora of different discrete actions that all converge to the same continuum action in an appropriate continuum limit. This is particularly troubling if one considers (this form of) discreteness to be fundamental. • Generically, discretisations break the fundamental diffeomorphism symmetry of general relativity [40–43]; in the discrete this symmetry is associated with an invariance under vertex translations [40–46]. As a consequence, former gauge degrees of freedom are ‘promoted’ to physical degrees of freedom and are a priori indistinguishable from the ‘true’ physical ones, thus called pseudo–gauge degrees of freedom [41, 42]. This can have serious repercussions e.g. in a path integral formulation, where a naive integration would lead to (additional) divergences once the diffeomorphism symmetry is restored in a continuum limit. • An issue related to the breaking of diffeomorphism symmetry is the (unphysical) dependence of the theory on the choice of the discretisation / regulator. There exist several proposals to remove this regulator, where it is a priori unclear which one is to be preferred. Should one some over all possible triangulations as in spin foam models and causal dynamical triangulations, or also all topologies as suggested by group field theory? Or should one rather consider a refinement limit, in which the theory is defined on finer and finer triangulations? Under certain conditions in spin foam models, summing over all foams and refining a single foam 2

4

Even though loop quantum gravity uses graphs to define (kinematical) states, it is a continuum theory: two states defined on two different graphs can be compared by embedding them into a common refinement, i.e. a graph, which is a refinement of both previous graphs. For this to work consistently, the embedding maps have to satisfy cylindrical consistency conditions. In an inductive / projective limit, one can then construct the continuum Hilbert space [20, 34, 35].

1.1 Non–perturbative quantum gravity may be identical [47]. In a canonical formulation, for broken diffeomorphism symmetry, the constraints of the theory rather occur as pseudo–constraints [41, 42, 48–50]. • Covariant approaches to quantum gravity have the goal to give meaning to the (formal) gravitational path integral Z ∂M

Dg exp{i SE–H [g]}

,

(1.1.1)

where Dg is the measure on the space of geometries, i.e. the space of metrics modulo diffeomorphisms. In the discrete setting, one not only has to replace the Einstein–Hilbert action SE–H [g] by a discrete action, but also choose a measure on the set of discrete geometries, which will directly translate into a choice on the measure on the space of geometries. Since diffeomorphism symmetry is generically broken, this raises the question whether the measure factor is anomaly free with respect to diffeomorphisms [51]. Additionally, choices on the measure in spin foam models influence the divergent behaviour of the model [46, 52–55]. • Lorentz invariance is generically broken by the introduction of a discretisation, e.g. in Regge calculus [17], yet its fate in quantum gravity is not clear. It has been frequently argued that quantum gravity effects at the Planck scale may result in a breaking or modification / deformation of Lorentz symmetry, which could be potentially observed as modified dispersion relations of matter propagating on this quantum gravitational background. A possible experimental test could be by observing highly energetic particles, which travelled over long, i.e. cosmological, distances, e.g. emitted in a gamma ray burst [56, 57]. The various issues stated above are very much related to one another and are rooted in the lack of good understanding of the relation between the discrete and the continuous description of gravity, which even arises on the classical level. This particularly affects the fate of diffeomorphism symmetry in the discrete setting. See also [58] for a review on the interplay between discretizations, diffeomorphism symmetry and constraints in various models of quantum gravity and how discretizations can be constructed to circumvent some of the issues. However, exactly this understanding is crucial in making the case for any of the discrete quantum gravity approaches above. Due to the lack of observational evidence and data indicating quantum gravity effects, the least a quantum gravity theory must achieve is consistency with the classical theory it intends to generalize. This can already be explored on the discrete level, where one would compare the candidate theory to discrete gravity, e.g. Regge calculus [17], yet the full consistency can only be checked for a continuum formulation. Hence the overarching question is, whether quantum gravity models defined in the discrete possess a phase, in which the collective dynamics of many building blocks exhibits a smooth description consistent with general relativity. This consistency certainly should also include (a realization of) diffeomorphism symmetry. In case such a scenario is realized one might be able to predict quantum gravity corrections to general relativity that motivate new experiments and tests, which might eventually verify or falsify the theory.

1.1.1 Introduction to Regge calculus To get a better understanding of this, let us concretely discuss a discretisation of classical general relativity, namely Regge calculus [17, 59]. Regge calculus is defined on a triangulation of a d– dimensional manifold, where the geometric information is encoded into the distance between the vertices, i.e. the length of the edges of the triangulation. Thus the edge lengths are the dynamical variables of the theory. Note that this formulation is inherently coordinate independent, since only the relative distances between the vertices determine the theory. The Regge action SR is given as follows:     X X (σd ) X X (σd ) SR := − Vh 2π − θh  − Vh π − θh  , (1.1.2) h⊂bulk

σ d ⊃h

h⊂bdry

σ d ⊃h

5

1 Introduction where σ d denotes a d–simplex and Vh is the volume of a (d − 2)–simplex, also called a ‘hinge’. The (σ d )

d–dim. (interior) dihedral angle of the d–simplex σ d located at the hinge h is denoted by θh and is determined by the edge lengths of the simplex. The terms in brackets resemble the bulk and boundary deficit angles respectively: (bulk)

ωh

(bdry)

ωh

:= 2π − := kπ −

X

(σ d )

θh

,

(1.1.3)

,

(1.1.4)

σ d 3h

X

(σ d )

θh

σ d 3h

where k depends on the number of pieces one is glueing together at this boundary. If there are (bulk) only two pieces we have k = 1. The deficit angles ωh define a notion of distributional curvature in Regge calculus: If the dihedral angles located at a hinge shared by several simplices do not sum up to 2π, then the geometry is locally curved. Since the edge lengths are the dynamical variables of Regge calculus, one obtains an equation of motion for each bulk edge by varying the action with respect to this edge length le 3 : X ∂Vh ∂SR =− ωh = 0 ∂le ∂le

.

(1.1.5)

h⊃e

To provide an intuitive example, in 3D, this formula simplifies notably because the edges are the hinges of the triangulation. Each bulk edge carries one bulk deficit angle, for which the equations of motion state that it has to vanish. Hence 3D Regge calculus is locally flat, which is consistent with the 3D continuous theory: Regge calculus ‘perfectly’ matches a flat geometry by glueing intrinsically flat simplices in a flat way, i.e. with vanishing deficit angles. In fact, this example can help us to better understand the issues mentioned above, since it illustrates a particular discretisation with an intact diffeomorphism symmetry in the discrete, i.e. a vertex translation invariance [40–46]. Another example is 3D Regge calculus with a cosmological constant [42,60], where the flat tetrahedra are replaced by constantly curved ones. Similar examples also exist in non–topological theories, for instance in 4D Regge calculus, in case the boundary data allow for a flat solution in the bulk [44, 45]. The key point in these examples is that the continuum dynamics are ‘perfectly’ represented in the discretisation; these approaches are thus called ‘perfect discretisations’ [61–63]. In more general situations, their construction becomes highly non–trivial: In order to pull back the continuum dynamics onto the discretisation, one first has to solve the continuum dynamics. On the other hand, one can observe that the symmetry is ‘less’ broken the finer the discretisation is. Thus one can attempt to improve the discretisation [42, 64, 65] by combining, i.e. coarse graining, building blocks of the discretisation into a new (effective) building block, such that the dynamics are endowed onto the discretisation in steps and eventually, in the limit of infinite refinement, diffeomorphism symmetry gets restored. For interacting systems, such as quantum gravity, the perfect realization of a diffeomorphism symmetry in the discrete is highly non–trivial and may not be possible, however, a suitable coarse graining algorithm should be able to at least approximately restore diffeomorphism symmetry. The hope is that given such a good approximation one can draw conclusions on the issues explained above and eventually uncover the relation of the (improved) quantum gravity model and general relativity. Indeed, building this connection between discrete quantum gravity models and general relativity is the main topic of this thesis, which will be mainly discussed for spin foam models. Therefore we will present a brief introduction into spin foams in the next section with particular focus on the current status of these models with respect to the issues discussed above. 3

6

Note the Schl¨ afli identity:

P

h⊂σ d

(σ d )

Vh δθh

= 0.

1.2 Spin foam models

1.2 Spin foam models The modern spin foam models are usually constructed as follows. Consider the Plebanski formulation of gravity [66]: Z SPleb = B ∧ F (A) + φB ∧ B , (1.2.1) M

where B is a Lie–algebra valued 2–form and F is the curvature 2–form of the gauge connection (1–form) A. If we just consider the first part of this action, i.e. B ∧ F (A), this gives a topological theory, called BF theory [67]. Topological means that there are no local degrees of freedom and the global ones are completely fixed by the topology of the manifold M 4 . In order to get a theory of gravity, constraints have to be imposed that break the (too many) symmetries of this action. These constraints are called simplicity constraints and are imposed via the Lagrange multipliers φ; if satisfied one returns to Palatini’s first order formulation of gravity. To construct a spin foam model, one usually starts with topological BF theory, for which it is rather well–known how to discretize and quantize it [67, 70], see also [46, 71, 72] for discussions on the issues in this construction. In many cases the chosen discretization is (dual to) a triangulation, but it can be generalized to arbitrary 2–complexes, i.e. a collection of vertices, edges and faces. Then, in order to break the symmetries of BF theory, one has to impose simplicity constraints in the discrete, quantized theory: the question, which constraints to impose and how is the key difference between the currently proposed spin foam models for 4D gravity, see e.g. the Barrett– Crane model [73], the EPRL–model [74] and the FK–model [75]. We would also like to mention the KKL–model [76], which generalizes the EPRL–model to arbitrary 2–complexes. In general, the dynamical ingredients of spin foam models can be summarized as follows: Spin foam models defined on 2–complexes assign an amplitude to this complex, if it connects e.g. two (discrete) 3–geometries on its boundaries, then this amplitude can be interpreted as a transition amplitude. The 2–complex itself is coloured with additional data: the faces carry irreducible representations of the symmetry group the spin foam is defined on, e.g. SO(4) in 4D Riemannian gravity or SL(2, C) in Lorentzian 4D gravity, whereas the edges carry projectors onto (a subspace of) the invariant subspace of the tensor product of irreducible representations meeting at this edge. Which subspace the projectors project on depends on the imposition of the simplicity constraints. If no constraints are imposed at all, we have the usual Haar projectors on the edges, which project onto the full invariant subspace. Then this prescription coincides with the strong coupling expansion of lattice gauge theories5 . Thus spin foam models can be regarded as generalized lattice gauge theories, which is nicely explained in the holonomy representation of spin foam models [69]. The entire amplitude of the complex (or foam) is then computed by summing over all the colourings on the faces6 . The overlying issue of spin foam models are their complexity, which greatly hinders the exploration of their dynamics. In fact, most of our knowledge and confidence for the validity of spin foam models comes from calculations involving only one simplex, i.e. one of the basic building blocks: A highly celebrated result is the asymptotic expansion of the vertex–amplitude, essentially the amplitude assigned to (the dual of) a simplex [23–29]. By scaling up the areas (of the triangles) of this simplex, which is frequently interpreted as a semi–classical limit, one can identify 4

R The Plebanski formulation of gravity is strikingly similar to the action of Yang–Mills theory: SYM = B ∧ F + g 2 B ∧ (?B) [68]. This action explicitly depends on a background, encoded in the Hodge dual ?. As one can see, if g → 0 one obtains BF theory as the weak coupling limit of Yang–Mills theory. We will argue below that one can understand spin foam models as generalized lattice gauge theories [69]. 5 This formulation can be obtained by performing a group Fourier transform of the functions associated to the faces / plaquettes of the lattice. 6 A prescription more commonly used in the literature is by directly assigning amplitudes to the vertices, edges and faces of the foam. This prescription is equivalent to the one discussed here. The vertex amplitude is directly constructed by a contraction of intertwiners, i.e. the basis elements of the invariant subspace, associated to the edges meeting at the vertex. See the review [77] for a nice explanation.

7

1 Introduction the dominating phase of the amplitude by a stationary phase approximation. This dominating phase (in the geometric sector) turns out to be the Regge action [17] associated to the simplex. Hence for the simplest possible triangulation, one obtains discrete gravity (in the geometric sector of) spin foam models in this particular limit. Note also that the amplitude is proportional to cos(SR ) = 12 (eiSR + e−iSR ) instead of just eiSR as for ‘ordinary’ path integrals. This is due to the fact that in a spin foam model one has to sum over both orientations of the simplex7 . Unfortunately, similar results relating spin foams and discrete gravity are rare for larger triangulations / foams, not to mention the continuum theory. In [78], the asymptotic geometry of arbitrary triangulations in the large spin limit of the boundary variables has been examined (without additional assumptions on the bulk triangulation), where one found that the amplitude is suppressed unless the geometry in the bulk satisfies accidental curvature constraints. This indicates that the large spin limit by itself might not be sufficient to explore the semi–classical geometry of spin foams, but also a refinement of the bulk (and the boundary) might be necessary, see e.g. [79]. Additionally even the asymptotics of one vertex amplitude are not completely understood, e.g. the measure factor, which is a possible indicator for the measure on the space of geometries chosen in spin foam models, is unknown. The work bundled in this thesis can be generally seen as the attempt to extend and deepen our understanding of the connection between spin foams and discrete / continuous general relativity. We explore this relation at very diverse numbers of involved building blocks, with particular focus on dynamical principles either to determine the theory or to improve the theory / discretisation, e.g. via coarse graining. At the lowest level, namely a single simplex / vertex amplitude, we revisit the asymptotic expansion of vertex amplitudes of the Ponzano–Regge model (in 3D) [23, 80] and the Barrett–Crane model (in 4D) [73] in chapters 2 / [30] and 3 / [31] with particular focus on deriving the measure factor from a coherent state approach (see [81] for a nice introduction); the result in [31] for the Barrett–Crane model is indeed the first of its kind. Starting from a conjecture derived in [43] relating discrete diffeomorphism invariance to triangulation independence, we investigate whether triangulation independence can be used as a dynamical principle to fix ambiguities in discrete gravity theories, e.g. define the path integral measure unambiguously. Invoking the relation between spin foams and Regge calculus from asymptotic expansions of vertex amplitudes, we study invariance of the (linearized) Regge path integral under local changes of the triangulation, so–called Pachner moves [82, 83], in chapters 4 / [59] and 5 / [84]. In 4D such a measure can almost be constructed, but turns out to be non–local, motivating more general coarse graining schemes that also affect the boundary of the discretisation. Such methods were invented in condensed matter theory, e.g. tensor network renormalization [85,86], in order to study effective dynamics of many degrees of freedom by only truncating the least relevant degrees of freedom. We apply these methods to (analogue) spin foam models (defined on quantum groups) in chapter 6 / [87] and uncover a rich fixed point structure with extended phases and phase transitions. Remarkably, the truncation scheme can also be understood as a tool to dynamically relate Hilbert spaces associated to fine or coarse boundaries. In chapter 7 / [88] we argue that such dynamical embedding maps can be interpreted as time evolution maps that allow us to define a consistent continuum theory from the discrete one, if these embedding maps satisfy so–called cylindrical consistency conditions [20, 34, 35, 89]. Before including the manuscripts in the order described above, we would like to give a brief introduction to them and elaborate on their context to the overarching question of this thesis.

7

8

This can be interpreted as a feature instead of an issue, since one can argue that in a gravitational path integral one has to integrate over all values of lapse and shift in order to impose the constraints of the theory, which can be seen as an evolution ‘back and forth in time’.

1.3 Overview of presented manuscripts

1.3 Overview of presented manuscripts 1.3.1 Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions In chapter 2, we present the paper [30], which directly deals with the question of how to compute asymptotic expansions of spin foam models, to be more precise the vertex amplitude / the amplitude associated to a simplex. In this particular paper we revisit the Ponzano–Regge model [23, 80], a theory for 3D (Riemannian) quantum gravity defined on a triangulation. The Ponzano–Regge model is actually the very first spin foam model. Its partition function Z is defined as follows: X Y Y Y j1 j2 j3  2j j1 +j2 +j3 , (1.3.1) Z= (−1) (2j + 1) (−1) j4 j5 j6 j

edges

triangles

tetrahedra

where ji denote spins, i.e. irreducible representations of SU(2), which are attached to the edges of the triangulation. This can be understood as assigning a length to the edge. In the partition function then, each edge is weighted with the dimension (of the vector space) of the representation j, whereas each tetrahedron is weighted by the 6j symbol made up of the six spins assigned to the six edges of the tetrahedron. One way to check the relation between spin foam models and classical (discrete) gravity is to extract a semi–classical geometry and action via an asymptotic expansion from the spin foam amplitude. To do so one identifies the dominating phase of the spin foam, which is then usually interpreted as the effective action of this system. One method to achieve this, which proved to be very useful in the examination of modern spin foam models [28, 29], is a coherent state approach. This concept is nicely explained in [81], here we will briefly discuss the idea in 3D: In a given representation j of SU(2) coherent states [90] are labelled by a vector ~n ∈ S2 , denoted as |αj (~n)i. This state is defined up to a phase by the following equations: D

~ j αj (~n) = i j αj (~n) ~n · L  m E ~ m αj (~n) L αj (~n) = i j (~n) j

,

(1.3.2)

.

(1.3.3)

~ j denotes the generator of rotations, represented in the representation j. The first equation states L that αj (~n) is an eigenstate of the rotation operator pointing in direction ~n, whereas the second one ~ j , which basically is the m component of ~n gives the expectation value of the m component of L times the spin j. In short, the coherent state is peaked on the vector pointing in the direction ~n. From these coherent states one can construct so–called coherent intertwiners, i.e. basis elements in the invariant subspace of the vector space of the tensor product of irreducible representations, by taking the group averaged tensor product of several coherent states. For a 3–valent intertwiner, basically a Clebsch–Gordan coefficient, this is a tensor product of three coherent states, which can be regarded as a ‘quantized triangle’. The vector associated to the coherent state is then interpreted as the edge vector. Remarkably, the amplitude for a Clebsch–Gordan coefficient is dominated by P3 geometries for which i=1 ji~ni = 0, which is known as the closure constraint. Such a triangle is also referred to as a coherent triangle, since it is peaked on a classical geometry. From these basic intertwiners, one can construct larger spin networks by contracting these intertwiners according to the combinatorics of the network. One particular example is the 6j symbol, the central building block of the Ponzano–Regge model, which is a contraction of four 3–valent intertwiners, i.e. a distinct glueing of four triangles forming a tetrahedron. Using the coherent states defined above, one performs a stationary phase approximation of the group integrations by uniformly scaling up the spins on the edges; the tetrahedron is ‘blown up’ to a macroscopic size. Remarkably, the stationary point conditions allow for a clear geometric interpretation. First, all

9

1 Introduction four triangles have to close, i.e. their edge vectors have to sum to zero. Second, one can define normal vectors to the triangles, which have to sum to zero as well, i.e. the tetrahedron constructed from the triangles has to close, too. The angles enclosed by the normal vectors of the triangles are the (exterior) dihedral angles of this tetrahedron. Eventually, one finds that the dominating phase of the 6j symbol is given by the Regge action [17] associated to the tetrahedron, a result known for about 45 years [23] and derived in a plethora of ways [91–95]. However, despite the nice geometric interpretation the coherent state approach generically fails to produce the complete asymptotic expansion. While it is straightforward to obtain the dominating phase via a stationary phase approximation, the latter cannot be completed, since the determinant of the Hessian matrix, sometimes called the measure factor8 , cannot be computed. This also troubles the calculations for 4D spin foam models, where the measure factor is generically unknown. Furthermore, even for the 6j symbol, where the measure factor is known, the coherent state approach fails and only numerical results are available [96]. To overcome this drawback, we introduce modified coherent states in [30], see also chapter 2, which allow for the same geometric interpretation as the usual ones, yet come with an additional smearing angle φ. The major advantage of these coherent states is that the stationary phase approximation can be performed in steps, first with respect to the smearing angles and then with respect to the group elements. The new effective action obtained after the first one turns out to be the first order Regge action [97], which already establishes the connection to discrete gravity and actually is the first derivation of this action from the asymptotics of spin foam models. Given the geometric interpretation of Regge calculus, we complete the asymptotic expansion by deriving several identities from curved Regge calculus [98] and derive properties of higher order corrections [99–102]. The main purpose of this work has been to develop important tools and ideas that might help in also extracting measure factors for 4D spin foam models. Indeed, we succeeded in this endeavour, as we will explain in the next section.

1.3.2 The Barrett-Crane model: asymptotic measure factor The paper [31] presented in chapter 3 can be seen as a direct continuation of [30] in chapter 2 and deals with the asymptotic expansion of the vertex amplitude of the so–called Barrett–Crane model, i.e. the amplitude assigned to a 4–simplex of the triangulation. The Barrett–Crane model is the very first, and arguably the simplest, spin foam model for 4D (Riemannian) gravity. In fact, the construction of the modern spin foam models is very similar as it is nicely illustrated in [69]: Starting from discretized and quantized BF theory, they differ in the implementation (and choice) of the simplicity constraints. In case of the Barrett–Crane model, these constraints are imposed ‘strongly’, such that they are generically satisfied. Indeed, this choice is very constricting, which is reflected in the fact that the 4–valent intertwiner (also called Barrett–Crane intertwiner), dual to a tetrahedron, is unique [73]. Thus, the Barrett–Crane model has no intertwiner degrees of freedom, in contrast to the other 4D spin foam models. Actually, the Barrett–Crane model is nowadays commonly disregarded as a viable theory of quantum gravity, for a recent discussion see [103]. One of the issues is the presence of non–geometric sectors, for example a BF sector, in the asymptotic expansion, i.e. the semi–classical limit of the model contains contributions that do not allow for a geometric (gravitational) interpretation. In fact this is also a feature shared by the other spin foam models. Additionally, the Barrett–Crane model lacks glueing constraints, that is the tetrahedra along which the simplices are glued together are made up of triangles with the same areas, yet the shapes of these triangles are not matching. Thus, in a semi–classical limit, the Barrett–Crane model rather gives area Regge calculus [104], a theory in which areas, instead of edge lengths, are the dynamical variables. This theory is troubled 8

This term is not to be confused with ‘measure’ in spin foam literature, which refers to edge and face amplitudes.

10

1.3 Overview of presented manuscripts by vanishing deficit angles, such that it is locally flat, and, due to the non–matching of the shapes of the faces, the metric, locally defined for each 4–simplex, is discontinuous [105, 106]. Despite the comparable simplicity of the Barrett–Crane model to the other 4D spin foam models, the asymptotic expansion of the vertex amplitude, i.e. the 10j symbol, is still incomplete, even when only the geometric contributions are considered. Remarkably, starting from a nice identity of the 10j symbol derived in [27], the tools and identities for Regge calculus developed in [30], see also chapter 2, can be almost analogously applied to the Barrett–Crane vertex amplitude. Surprisingly, the result, which is the first of its kind, has a nice geometric interpretation and is almost completely explicit; the only implicit contribution is a Jacobian from a variable transformation from areas to edge lengths. Indeed, this Jacobian can be absorbed by considering a peculiar triangulation. Instead of examining only one 4–simplex, we discuss two 4–simplices glued together along all of their tetrahedra, which is the simplest possible triangulation of a 4–sphere. Then the Jacobian can be absorbed by introducing shape–matching constraints, i.e. constraints that enforce that the areas (triangles) are constructed from edge lengths and thus have matching shapes.

1.3.3 Path integral measure and triangulation independence in discrete gravity Chapter 4 includes paper [59], in which we address the choice of the path integral measure in (linearized) Regge calculus. Regge calculus [17, 18] is a discretization of classical general relativity defined on a triangulation, see also chapter 1.1.1 for a brief introduction. Asymptotic expansions in spin foam models indicate, as discussed above, that one obtains Regge gravity in a semi–classical limit. Since the measure factors for spin foam models are hardly known, one can invert the logic and ask, whether one can derive and define a suitable measure factor for quantum Regge calculus, which then presents a ‘blueprint’ spin foams can be compared to. Indeed, many different measure factors have been proposed for Regge calculus [107–109], yet none has been derived from a dynamical principle. In chapter 4, see also [59], we propose to use triangulation independence as the determining property for the measure. This is motivated by the work [43], in which perfect discretisations [41, 42, 61, 62], i.e. discretisations with intact discrete diffeomorphism symmetry, have been constructed using a coarse graining scheme. There it has been conjectured that having a discrete diffeomorphism invariant theory is equivalent to the fact that this theory is triangulation / discretisation independent. This conjecture is explained in more detail in the discussion (see chapter 8.2). Let us emphasize that this is a desirable feature: A theory being both discretisation independent and equipped with a discrete (remnant of) diffeomorphism symmetry fixes many ambiguities in the choices of possible discretisations. Additionally, due to discretisation independence the continuum limit can be performed trivially, which allows to check consistency with the continuum theory, while calculations, such as expectation values of observables, can be performed in the discrete, even on the simplest triangulation possible. Most importantly, the discrete version of diffeomorphism symmetry ensures the realization of diffeomorphism symmetry in the continuum. Clearly, satisfying these conditions is highly non–trivial. Thus, instead of applying a full fledged coarse graining scheme to (quantum) Regge calculus, we approach the problem from the opposite direction, namely triangulation independence. For a triangulation, this can be tackled on a local level: A triangulation of a manifold can be converted into any other triangulation of the same manifold by a consecutive application of Pachner moves [82, 83], i.e. local changes of the triangulation. A theory is triangulation independent, if it gives the same result for any triangulation of the same manifold, which is equivalent to invariance of the theory under Pachner moves. On the classical level, it is well–known that the discrete Regge action is fully triangulation invariant in 3D, where gravity is topological, and almost invariant in 4D [59]. Hence we examine, whether one can derive a measure for quantum Regge calculus that renders the path integral triangulation independent, i.e. invariant under Pachner moves. Instead of discussing the full

11

1 Introduction theory, we instead perturb the Regge action around a flat background solution, i.e. vanishing deficit angles, up to quadratic order and perform the path integral for the perturbations. The resulting path integral measure only depends on the background variables. To sum up, in chapter 4, see also [59], we use triangulation independence, i.e. invariance under Pachner moves, as a dynamical principle to construct the path integral measure in quantum Regge calculus. In 3D, this construction is successful and consistent with the asymptotics of the Ponzano– Regge model [23]. However in 4D, already the Regge action is not invariant under all Pachner moves and the construction of a triangulation invariant measure (for the remaining moves) is complicated by the appearance of an overall factor in the Hessian matrix, which is the topic of the paper briefly introduced in the next section.

1.3.4 Discretization independence implies non–locality in 4D discrete quantum gravity The work [84] presented in chapter 5 is the direct continuation of the attempt in [59], see also chapter 4, to construct a triangulation invariant path integral measure in 4D (linearized) Regge calculus. While the calculation is strikingly similar to the 3D case, the Hessian matrix in 4D additionally possesses an overall factor, which resisted a geometric interpretation and has been conjectured in [59] to be non–local. Indeed this factor troubles the construction of a triangulation invariant measure in 4D, such that the purpose of the paper [84] is to derive its geometric meaning and actually prove its non–local nature. Interestingly, this overall factor turns out to be a criterion determining whether the six vertices involved in the 4D Pachner move lie on the same 3–sphere [110], see also [111]; if this is the case, the factor vanishes. Therefore we carefully analyse the situations, in which this can happen, which includes a careful examination of the relative orientation of simplices in simplicial complexes, see also [112]. Indeed, we identify configurations in which the factor vanishes, yet all involved simplices are non–degenerate. Thus we show that the factor necessarily is non–local and cannot be written as a product of amplitudes associated to (sub)simplices. The result is surprising since one might have hoped for the existence of a topological sector in 4D gravity, which describes flat geometries. Whereas this is the case for the classical theory, it does not exist in the quantum theory. In fact, non–localities are expected to appear under coarse graining of interacting theories, which generically complicates the calculations and demands the implementation of approximations to keep the scheme feasible, e.g. the truncation scheme invented by Migdal [113] and Kadanoff [114] outright removes the non–local interactions, but is remarkably predicting a phase transition for the 2D Ising model. Yet the error made by such ad hoc truncations is difficult to estimate, such that more efficient truncation schemes are desirable, which are capable of treating the non–local excitations effectively, e.g. by also changing the boundary data in the process. Such coarse graining schemes, e.g. tensor network renormalization [85, 86], originally developed in condensed matter theory, are applied to (analogue) spin foam models defined on quantum groups in the paper [87] in chapter 6, which is briefly introduced in the next section.

1.3.5 Quantum group spin nets: refinement limit and relation to spin foams Up to this point of the thesis, we have not addressed the elephant in the room: how can we extract the collective dynamics of many building blocks of spin foam models? This questions is very difficult to answer due to the complexity of spin foam models. An idea to make progress is employing numerical simulations, yet quantum Monte Carlo simulations are already out of the game, since they cannot handle complex amplitudes appearing in spin foam models. Surprisingly the initial setup of spin foam models is not too different from the initial starting point in condensed matter theory: given a discretisation with (locally) interacting degrees of freedom placed e.g. on the vertices of the discretisation, one intends to extract the macroscopic behaviour

12

1.3 Overview of presented manuscripts from the many–body dynamics, e.g. ground states and the spectrum of the Hamiltonian. Tools to study effective dynamics are summarized under the term real space renormalization techniques [115], in contrast to momentum space renormalization in quantum field theories, of which many can be seen as coarse graining schemes, i.e. mechanisms to combine finer degrees of freedom into new, coarser ones. Generically, this leads to non–localities namely the coarser degrees of freedom start to interact (more) non–locally in comparison to the finer ones. However in the past twenty years many new renormalization schemes have been developed, which are much more suited to effectively deal with non–local interactions, e.g. density matrix renormalization group (DMRG) [116, 117], matrix product states (MPS) and Projected Entangled Pair States (PEPS) [118, 119] and tensor network renormalization [85, 86] in different flavours [120]. The idea of the latter is to encode the dynamics of the system into a tensor network, where each tensor carries the dynamics of the local degrees of freedom. Then the partition function is rewritten as a contraction of this tensor network. Since this is only a rewriting of the initial problem, tensor network renormalization attempts to approximate the partition function of the original system by a contraction of a coarser network by (locally) combining tensors into a new, locally interacting tensor. However under consecutive application of this coarse graining method, the boundary data of the tensor grow exponentially, such that one has to employ truncations. In order to still provide a good approximation to the initial system, it is thus vital to identify the relevant degrees of freedom: this is achieved by employing a singular value decomposition (of a recombination of the tensors), which transforms the fine degrees of freedom into an orthogonal basis, ordered according to their relevance indicated by the associated singular value. On the one hand this particular variable transformation allows to employ good approximations by only keeping the most relevant degrees of freedom, i.e. with the largest singular values, while on the other hand it serves as a direct translation between the degrees of freedom on the fine and the coarser discretisation. If interpreted the other way around, this variable transformation can be seen as an embedding map, computed directly from the dynamical ingredients of the system, embedding the coarse boundary data into the fine data. To be more precise, such an embedding map relates the Hilbert spaces associated to the coarser and finer boundaries. This is a crucial feature which partially motivated the conceptual work [88] in chapter 7, in which we describe how to construct a consistent continuum theory using these embedding maps. Remarkably, the dynamical ingredients of spin foam models, i.e. the projectors (onto invariant subspaces of the tensor product of irreducible representations) located on the edges, fit very well into the language of tensors. For each face meeting the edge, the tensor carries an index, which runs over all labellings assigned to the pair (edge, face)9 , usually representation labels (magnetic indices). Instead of coarse graining such networks, which is currently work in progress, we consider simplified versions, so–called spin nets, in [87], see also [39]. Spin nets are dimensionally reduced spin foams, so instead of a 2–complex they are defined on a 1–complex, a graph, yet we keep the main dynamical ingredients of spin foams, i.e. the projectors, which are now located on vertices. In fact, one can assign the entire dynamical ingredients to the vertices of the lattice, such that these models are also known as vertex models. Then the corresponding tensor network is straightforward to construct and consists only of one type of tensors. Crucially, the projectors permit an interpretation in terms of simplicity constraints. A second simplification is the replacement of the underlying Lie group e.g. by a finite group, in order to only have finitely many representation labels to sum over in the partition function, which allows for efficient numerical simulations. In [87], see also chapter 6, we instead define spin nets on the quantum groups SU(2)k , which on the one hand also contains a natural cut–off on the representation labels (depending on the level k), yet on the other hand is the full ‘group’10 used 9

To write a spin foam as a tensor network, one has to additionally introduce auxiliary tensors on the faces that ensure that the face carries the same irreducible representation [37]. This is also necessary in order to rewrite lattice gauge theories as tensor networks [121]. 10 Despite their name, quantum groups are no groups, but quasi-triangular Hopf algebras [122, 123]. Here we

13

1 Introduction in spin foam models describing gravity with a non–vanishing cosmological constant. This intuition originates from the Turaev–Viro spin foam model [124] describing 3D quantum gravity with a non– vanishing, positive cosmological constant, which essentially resembles a q–deformed version of the Ponzano–Regge model. In recent years it has also been explored to define loop quantum gravity with a non–vanishing cosmological constant, in particular in (2+1)D to establish the connection to the Turaev–Viro model [125–129], and similarly to extend 4D spin foam models to quantum groups [130–132]. To these quantum group spin nets we apply a symmetry preserving version of tensor network renormalization, see also [39] for this method, to examine their many–body dynamics. We put special focus on the projectors, now located on the vertices: In the modern spin foam model construction, these are not projectors onto the full invariant subspace (of the tensor product of irreducible representations), but only a smaller subspace via the implementation of simplicity constraints. The question is how these projectors change under coarse graining, or to put it differently, how the simplicity constraints change: does the additional structure, added by allowing smaller invariant subspaces, survive under coarse graining or does the system flow back to the standard (analogue) lattice gauge theory phases, namely either the weak coupling, ordered BF phase or the degenerate, strong coupling phase? In a refinement limit, reached once the system has arrived at a fixed point of the coarse graining procedure, we thus examine whether spin nets allow for more structure, e.g. more fixed points than lattice gauge theories or even extended phases (with phase transitions). Indeed we find a rich fixed point structure beyond the standard phases equipped with extended phases in parameter space with phase transitions that show tentative indications to be of second order. This is very encouraging evidence that also spin foam models possess more structure than standard lattice gauge theories and may potentially have a phase with effective dynamics consistent with general relativity.

1.3.6 Time evolution as refining, coarse graining and entangling As briefly mentioned in the previous section, tensor network renormalization provides a scheme to compute embedding maps from the dynamics of the theory relating fine to coarse boundary data. Such dynamical embedding maps possess a greater potential than merely providing a truncation scheme under coarse graining: If we use them to refine the discretisations, we should eventually arrive at a continuum description from the discrete theory. To improve our understanding in this direction, we tackle the conceptual issues in defining a continuum theory from a discrete one in [88], see also chapter 7. As the title of the paper suggests, the key idea is to understand time evolution in discrete systems as a map that relates different discretisations, in particular discretisations that are variably coarse (or fine), i.e. discretisations that are capable to capture differing numbers of degrees of freedom. This idea is in fact motivated from local time evolution moves in (canonical) Regge calculus [133–135], which time evolve a triangulated hypersurface to another one, where the latter can capture more, less or the same number of degrees of freedom, possibly organized in a different way. On the classical level, this is captured by phase spaces of differing dimension associated to the hypersurfaces. Remarkably, if the number of degrees of freedom change under local time evolution, this always results in the appearance of constraints, even if one does not model gravity or diffeomorphism symmetry is broken. The occurring constraints are called pre– and post–constraints: the pre–constraints are conditions that have to be satisfied in order for time evolution to take place, while the post–constraints are automatically satisfied once time evolution has taken place. Notably for gravity, in the example of a refining time evolution, the post–constraints also contain diffeomorphism and Hamiltonian constraints. In a quantum theory one assigns Hilbert spaces to the triangulated hypersurfaces. understand SU(2)k as the q-deformation of the universal enveloping algebra of SU(2), denoted as Uq (su(2)). π q = exp{i k+2 } is a root of unity, k denotes the level of the quantum group.

14

1.3 Overview of presented manuscripts Then local time evolution acts as an embedding map of the initial into a different Hilbert space. See also [136, 137] for discussions of the quantum case. The observant reader may immediately object that time evolution generated by local moves may depend on the particular choice and order of the moves, which may render this approach to be unphysical. Indeed, to avoid such pathologies, time–evolution has to be path–independent [138], which translates into consistency conditions called (dynamical) cylindrical consistency for the embedding maps, a concept well–known (on the kinematical level) in loop quantum gravity [20, 34, 35]. The extension of the concept towards dynamical embedding maps has already been explored in [89], a viewpoint we extend upon in chapter 7, see also [88]. The important feature of dynamical cylindrical consistency is that, if realized, states represented on different discretisations can be compared to one another and identified as describing the same physical situation, i.e. physics should not depend on which discretisation it is represented on. As a consequence, the state of the system only implicitly depends on the specific discretisation, in fact it can already be represented in the continuum. Furthermore, in gravity the interpretation of embedding maps as time evolution is particularly appealing, since in gravity, as a totally constrained system, time evolution is a gauge transformation. Thus, the physical state of the system does not change under time evolution; indeed if cylindrical consistency is satisfied, time evolving the state means embedding it into a different discretisation, which by definition lies in the same equivalence class. The physical state therefore does not change, it is simply expressed on a different discretisation. Candidate’s contribution to the presented manuscripts Before including the manuscripts representing the bulk of this thesis, we have to clarify and outline the candidate’s contribution to the multi–authored papers: In the projects concerning the asymptotic expansions of vertex amplitudes in spin foam models [30, 31] the author of this thesis was involved in all major steps, yet his main contribution was for the calculations on the partial approximations of stationary phase, i.e. first with respect to the smearing angles and afterwards for the dihedral angles. To complete the latter the candidate derived several non–trivial identities for first order Regge calculus crucial for the extraction of the asymptotic measure factor in both papers [30, 31]. Also, the calculations in [30] were quite involved: The candidate’s knowledge of numerical methods were crucial for finishing the project, even though they are not directly visible in the final version of the paper. The papers were written collaboratively, yet the candidate wrote significant portions of the introduction and discussion in both papers, and is mainly responsible for the organisation of the paper [30]. The papers [59, 84] concerning triangulation independence of the linearized Regge path integral were mainly developed and authored by the author of this thesis, of course with several discussions and consultations with the co–authors. The project on coarse graining quantum group spin nets [87] was a joint effort by all three authors. Again the candidate was involved in all main steps of the work, including the definition of the model and dealing with the intricacies of quantum groups in the development of the algorithm. Most of the candidate’s efforts were invested in coding and optimizing the algorithm, in particular the higher accuracy version, running of the simulations and analyzing the data. The paper was written collaboratively, where the candidate wrote the parts on diagrammatic calculus, the coarse graining algorithm and the results of the simulations. Finally, the conceptual paper on interpreting embedding maps as time evolution maps [88] was a result of long contemplation on fundamental and conceptual issues in quantum gravity. As a result the paper is mostly conceptual and many points arose in discussions between the candidate and Bianca Dittrich clarifying the direction and interpretation of our research. Concurring the explicit writing, the candidate contributed the sections on topological theories and the geometric interpretation.

15

2 Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions Wojciech Kami´ nski1,2,3 and Sebastian Steinhaus2,3 1

Wydzial Fizyki, Uniwersytet Warszawski Ho˙za 69, 00-681, Warsaw, Poland

2 Perimeter

Institute for Theoretical Physics 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5 3 Max

Planck Institute for Gravitational Physics Am M¨ uhlenberg 1, D-14476 Potsdam, Germany

published in: J. Math. Phys. 54 (2013) 121703, [arXiv:1307.5432 [math-ph]]. Copyright 2013 AIP Publishing LLC. http://dx.doi.org/10.1063/1.4849515

Abstract We present the first complete derivation of the well-known asymptotic expansion of the SU (2) 6j symbol using a coherent state approach, in particular we succeed in computing the determinant of the Hessian matrix. To do so, we smear the coherent states and perform a partial stationary point analysis with respect to the smearing parameters. This allows us to transform the variables from group elements to dihedral angles of a tetrahedron resulting in an effective action, which coincides with the action of first order Regge calculus associated to a tetrahedron. To perform the remaining stationary point analysis, we compute its Hessian matrix and obtain the correct measure factor. Furthermore, we expand the discussion of the asymptotic formula to next to leading order terms, prove some of their properties and derive a recursion relation for the full 6j symbol.

17

2 Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions

2.1 Introduction Spin foam models [A1–A4] are candidate models for quantum gravity invented as a generalization of Feynman diagrams to higher dimensional objects. Their popularity is rooted in the fact that they were well adapted to descibe 3D Quantum Gravity theories such as the Ponzano-Regge [A5, A6] or the Turaev-Viro model [A7]. To examine whether these models are a quantum theory of 4D General Relativity, in particular whether one obtains Gravity in a semi-classical limit is an active area of topical research. One of the strongest positive implications comes from the asymptotic analysis of single simplices in spin foam models: A first attempt to compute the asymptotic expansion of the amplitude associated to a 4-simplex in the Barrett-Crane model [A8] can be found in [A9,A10]. This was continued for the square of (the Euclidean and Lorentzian) 6j and 10j symbols in [A11], whereas the most recent asymptotic results for modern spin foam models, i.e. the EPRL-model [A12] or the FK-model [A13], were obtained using a coherent state approach [A14–A19]: The basic amplitudes of the spin foam model are their vertex amplitudes (SU (2) 6j symbols in the 3D Ponzano Regge model). They are defined in a representation theoretic way and can be constructed from coherent states of the underlying Lie group [A20] as a multidimensional integral to which the stationary point approximation is applicable [A21]. This method has proven to be very efficient in determining the dominating phase in the asymptotic formula as well as the geometric interpretation of the contributions to the asymptotic expansion in spin foam models [A14–A19]. In 3D, on the points of stationary phase, 6j symbols are geometrically interpreted as tetrahedra, their dominating phase given by the Regge action [A22, A23], a discrete version of General Relativity on a triangulation. Similar results were proven by this method for the 4-simplex [A9,A14–A18] in spin foam models. Until today, this is still one of the most promising evidences that spin foam models are viable Quantum Gravity theories. Despite this success, the coherent state approach fails to produce the full amplitude. It has not yet been possible to compute the so-called measure factor, a proportionality constant (depending on the representation labels) in the asymptotic expansion, which is given by the determinant of the matrix of second derivatives, i.e. the Hessian matrix, evaluated on the stationary point. This failure even applies to the simplest spin foam model in 3D, the Ponzano-Regge model [A6], whose vertex amplitude is the SU (2) 6j symbol. To the authors’ best knowledge the Ponzano and Regge formula [A5] has not yet been obtained this way; we can only refer to numerical results in [A19]. This is particularly troubling for the coherent state approach, since the full asymptotic formula for SU (2) 6j symbols introduced in [A5] has been proven in many different ways, for example, by geometric quantization [A24], Bohr-Sommerfeld approach [A25], Euler-MacLaurin approximation [A26] or the character integration method [A11]. The source of the problem is the size of the Hessian matrix and the lack of immediate geometric formulas for its determinant. For the 6j symbol, for example, this matrix is 9 dimensional and its entries are basis dependent. This is a major drawback of the coherent state approach, in particular, since the full expansion is necessary to discuss and examine the properties of spin foams models of Quantum Gravity. To obtain this measure factor and compare it to other approaches [A27, A28], the complete asymptotic expansion is indispensable. This is an important open issue for 4D spin foam models. Our approach to overcome this problem can be seen (as we will show in Appendix 2.H) as a combination of the coherent state approach [A14–A19] and the propagator kernel method [A29]. It inherits nice geometric properties from the coherent state analysis with a similar geometric interpretation of the points of stationary phase. Moreover, the Hessian matrix is always described in terms of geometrical quantities and, most importantly, its determinant can be computed for the 6j symbol. In addition to the computation of the asymptotic formula of the 6j symbol [A5], our approach allows us to propose a new way to compute higher order corrections to the asymptotic expansion. These corrections have already been discussed in [A30,A31]: it was conjectured that the asymptotic

18

2.1 Introduction expansion has an alternating form   X    X  π 1 π 1 θi + + A1 sin ji + θi + + ... {6j} = A0 cos ji + 2 4 2 4

,

(2.1.1)

where An are consecutive higher order corrections and homogeneous functions in j + 12 . Our method allows us to prove this conjecture to any order in the asymptotic expansion.

2.1.1 Coherent states and integration kernels The coherent state approach is based on the following principle: Invariants (under the action of the group) can be constructed by integration of a tensor product of vectors (living in the tensor product of vector spaces of irreducible representations) over the group, i.e. group averaging. Since the invariant subspace of the tensor product of three representations of SU (2) is one-dimensional, the invariant is uniquely defined up to normalization. However, in order to apply the stationary point analysis the vectors in the construction above cannot be chosen arbitrarily. The choice, from which the method takes its name, is the coherent states class, which consists of eigenvectors of the generators of rotations with highest eigenvalues [A20]. Although these states are very effective in obtaining the dominating phase of the amplitude, the associated Hessian matrix turns out to be very complicated. This problem occurs since the action is not purely imaginary, which is also related to the problem of choice of phase for the coherent states which has not yet been fully understood. Both latter problems disappear if, instead of eigenstates with maximal eigenvalues, we take null eigenvectors for a generator of rotations L. Since this vector is trivially invariant with respect to rotations generated by L, the phase problem disappears. Similarly the contraction of invariants can be expanded in terms of an action that actually is purely imaginary. There is a trade-off, though: The quantity of stationary points increases and their geometric interpretation becomes more complicated. Moreover, frequently there exist no such eigenvectors for certain representations, (half-integer spins for SU (2)) and their tensor product gives thus vanishing invariants. The solution to these issues comes from the simple observation that null eigenvectors can be obtained by the integration of a coherent state, pointing in direction perpendicular to the axis of L, over the rotations generated by L. Like that the geometric interpretation usually obtained when using coherent states is restored. Furthermore, if we first perform the partial stationary phase approximation with respect to the additional circle variables, we obtain a purely imaginary action. In the special case of the 6j symbol, our construction allows us to write the invariant purely in terms of edge lengths and dihedral angles of a tetrahedron, in particular we perform a variable transformation from group elements to dihedral angles of the tetrahedron. The resulting phase of the integral is given be the first order Regge action [A32].

2.1.2 Relation to discrete Gravity Regge calculus [A22, A23] is a discrete version of General Relativity defined upon a triangulation of the manifold. Influenced by Palatini’s formulation, a first order Regge calculus was derived in [A32], in which both edge lengths and dihedral angles are considered as independent variables and their respective equations of motion are first order differential equations. Additional constraints on the angles have to be imposed in order to reobtain their geometric interpretation. These consist of the vanishing of the angle Gram matrix that implies the existence of the flat n-simplex with the given angles. Our derivation of the Ponzano-Regge formula shows astonishing similarity to this procedure. Moreover, from our calculation one can deduce a suitable measure for first order (linearized) Quantum Regge calculus, such that the expected Ponzano-Regge factor √1V appears, which naturally leads to a triangulation invariant measure [A27].

19

2 Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions Another version of 4D Regge calculus was explored in [A33] with areas of triangles and (a class of) dihedral angles as fundamental and independent variables. Several local constraints guarantee that the geometry of a 4-simplex is uniquely determined. These variables were chosen in the pursuit to better understand the relation between discrete gravity and 4D spin foam models. The latter are based on a similar paradigm as the Ponzano-Regge or the Turaev-Viro models [A5, A7] in 3D, yet enhanced by the implementation of the simplicity constraints from the Plebanski formulation of General Relativity [A34]. Area-angle variables as a discretization of Plebanski rather than EinsteinHilbert formulation were conjectured to be more suitable to describe the semi-classical limit of those models. Although it is known that the asymptotic limit of the amplitude of a 4-simplex for 4D gravity models is proportional to the cosine of the Regge action [A15–A17], the proportionality factor still remains unknown. We hope that the method presented in this work can help in filling the gap.

2.1.3 Problem of the next to leading order (NLO) and complete asymptotic expansion The asymptotic expansion for the SU (2) 6j symbol, in particular for the next to leading order (NLO), is still a scarcely examined issue, since it is very non-trivial to write the (NLO) contributions in a compact form. Steps forward in this direction can be found in [A30,A31,A35], where the latter gives the complete expansion in the isosceles case of the 6j symbol. The stationary point analysis applied in this work allows for a natural extension in a Feynman diagrammatic approach. From this approach the full expansion can be computed in principle, however in a very lengthy way. We derive a recursion relations of the Ward-Takesaki type, which is surprisingly similar to the one invented in [A36, A37] however in very different context, that, basically can be used in the asymptotic expansion to derive the NLO in a more concise way. Moreover, we can show explicitly that the consecutive terms in the expansion (2.1.1) are of the conjectured ‘sin/cos’ form.

2.1.4 Organization of the paper This paper is organized as follows: In section 2.2 we will present our modified coherent states, how to use them to construct invariants and how to contract these invariants to compute spin network amplitudes. The contracted invariants will be used to define an action for the stationary point analysis, which will be examined whether it allows for the same geometric interpretation on its stationary points as other coherent state approaches [A14–A19]. Its symmetries as well as the group generated by the symmetry transformations will be discussed. Section 2.3 deals with the partial stationary point analysis with respect to the introduced circle variables. This will allow us to write the amplitude, after a variable transformation, purely in terms of angle variables, which will be identified as exterior dihedral angles of a polyhedron. In section 2.4 we focus on the example of the 6j symbol. After another variable transformation, we obtain the action of first order Regge calculus and perform the remaining stationary point analysis. Eventually we obtain the asymptotic formula from [A5]. In section 2.5 we prove the conjecture from [A30, A31] that the full asymptotic expansion is of alternating form (2.1.1) and derive the recursion relations for the full 6j symbol. We conclude with a discussion of the results and an outlook in section 2.6. We would like to point out that several results of this paper have been obtained by tedious calculations which we did not include in its main part to improve readability. Interested readers are welcome to look them up in the appendices.

20

2.2 Modified coherent states, spin-network evaluations and symmetries

2.2 Modified coherent states, spin-network evaluations and symmetries In this section we are going to present the modified coherent states, how to construct the spinnetwork evaluation from them and that they allow for the same geometric interpretation in the stationary point analysis as similar coherent state approaches. Furthermore the symmetries of the action will be investigated. Consider a three-valent spin network, i.e. a graph with three-valent nodes carrying SU (2) intertwiners and edges carrying irreducible representations of SU (2). For each edge of the spin network we introduce a (fiducial) orientation such that each node of the network can be denoted as the ‘source’ s(e) or the ‘target’ t(e) of the edge e. Later in this work we intend to give a geometrical meaning to the spin network, in terms of polyhedra, triangles, etc. so we denote the set of nodes by F and the set of edges by E, which will become the set of triangles / faces and set of edges of the triangulation respectively. This dual identification is not always possible but we restrict our attention to the case of planar (spherical) graphs, where such notions are natural.

2.2.1 Intertwiners from modified coherent states Intertwiners are invariant vectors (with respect to the action of the group) in the tensor product of vector spaces associated to irreducible representations of that group. In the case of 3 irreducible representations of SU (2) the space of invariants is one dimensional and, moreover, there is a unique choice for the invariant for a given cyclic order of representations [A38, A39]. Suppose ξ ∈ Vj1 ⊗ · · · ⊗ Vjn is a vector in the tensor product of vector spaces of representations, then Z dU U ξ (2.2.1) SU (2)

is invariant under the action of SU (2). If ξ is chosen in a clever way, such an invariant is nontrivial. In the case of three representations it must be proportional to the unique invariant defined in [A38, A39]. In the following we present a choice which has the advantage that the method of stationary phase can be directly applied. For every face f , which is bounded by three edges, we choose a cyclic order of these edges (jf e1 , jf e2 , jf e3 ), labelled by the carried representations. These choices influence the orientation of the spin network [A38,A39] and are used to define and determine the sign of its amplitude, see also appendix 2.A. We introduce the following intertwiners for every face f : Z Z Y Y
2j dφji Cf = dUf Uf ff ({φf e }e∈F ) Oφf e |1/2i e , (2.2.2) 2π SU (2) j

e∈F

where ff is a function of the three angles φf e , e ⊂ f , |1/2i is the basic state of the fundamental representation and Oφ is a rotation matrix on R2 :   cos φ sin φ Oφ = . (2.2.3) − sin φ cos φ As mentioned above, (2.2.2) is invariant under the action of SU (2). Before moving on, we would like to outline the key differences between the approach described above and the usual coherent state approach [A15–A17, A19]. • Coherent states of SU (2) are labelled by vectors in R3 . On the stationary point with satisfied reality conditions, one obtains the geometric interpretation that for every face these three vectors form the edge vectors of a triangle. Later on we will prove the same geometric interpretation for the invariant Cf .

21

2 Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions • Furthermore we smear the coherent state by a rotation, which is the key ingredient of our approach. In addition to the stationary point analysis with respect to the {Uf }, we will also perform a stationary point analysis for the smearing angles {φf e }. Clearly, this will result in more stationary points contributing to the final amplitude. To suppress their contributions, we introduce modifiers ff which will be described in the next section. Prescription of the modifiers In order to make (2.2.2) complete, we have to describe the function ff . P For every face f we choose three vectors ve (e ⊂ f ) on R2 with norms je such that e⊂f ve = 0, i.e. they form a triangle with edge lengths je . The vectors are ordered anti-clockwise, their choice is unique up to Euclidean transformations, i.e rotations and translations. Let us denote the edges (in cyclic order) by 1, 2, 3. The angles (counted clockwise) between the vectors vk and vj are denoted by 2(ψkj − π), where 2(ψkj − π) is the SO(3) angle taking values in (0, π) for (k, j) ∈ {(2, 1), (3, 2), (1, 3)}. Due to the ordering, the SU (2) angles ψ21 , ψ32 and ψ13

v2 v3

2(ψ12 -π)

f v1

Figure 2.1: The choice of vectors vi .

are positive and smaller than 2π, in fact, one can also check that ψkj is in (π, 2π). This choice contributes an overall sign to the invariant, to be more precise, there are two different choices of cyclic order giving two invariants that may differ by a sign factor. This will be discussed in more detail in appendix 2.A.2. In particular we compare them to the intertwiner introduced in [A38,A39]. The angles ψkj satisfy the relation ψ21 + ψ32 + ψ13 = 4π .

(2.2.4)

We introduce a function f (x mod 2π, y mod 2π) such that • it is equal to 1 in the neighbourhood of x = ψ21 , y = ψ32 , • it is equal to zero in the neighbourhood of points (x, y) = ±(ψ21 + π, ψ32 ), ±(ψ21 , ψ32 + π), ±(ψ21 + π, ψ32 + π), (−ψ21 , −ψ32 )

.

(2.2.5)

Hence, we define ff (φf 1 , φf 2 , φf 3 ) = f (φf 2 − φf 1 , φf 3 − φf 2 )

.

(2.2.6)

The spin network evaluation Given the definition of invariants in (2.2.2) it is straightforward to define the evaluation of a given spin network: The intertwiners are contracted with each other according to the combinatorics of

22

2.2 Modified coherent states, spin-network evaluations and symmetries the network. The resulting amplitude has to be normalized, i.e. divided by the product of norms of our intertwiners, see section 2.3.4. It is, however, not sufficient in order to agree with the canonical definition [A38, A39]. The remaining sign ambiguity will be resolved in Appendix 2.A. As in the standard coherent state approach the amplitude (contraction of intertwiners) then reads: Z Y Y d φf e Y s (−1) d Uf ff ({φf e }e⊂f ) 2π f ∈F e⊂f f 2je Y  , (2.2.7)  Us(e) Oφs(e)e |1/2i , Ut(e) Oφt(e)e |1/2i e∈E

{z

|

}

eS

where s is the sign factor as prescribed in [A38, A39] (see Appendix 2.A), and (·, ·) is an invariant bilinear form defined by (|1/2i, |1/2i) = (| − 1/2i, | − 1/2i) = 0,

(|1/2i, | − 1/2i) = −(| − 1/2i, |1/2i) = 1 .

(2.2.8)

The choice of the orientation of edges, faces and the sign factor prescription will be described in appendix 2.A.1. To perform the stationary point analysis we rewrite (part of) the integral kernel as an exponential function and define the ‘action’ S. From (2.2.7) one can deduce that X S= Se , (2.2.9) e

where the action Se (labelled by the edge e) is given by:   Se = 2je ln  Us(e) Oφs(e)e |1/2i , Ut(e) Oφt(e)e |1/2i

.

(2.2.10)

2.2.2 The action In order to examine the geometric meaning of the action on its points of stationary phase, let us introduce the following geometric quantities. For each face f ∈ F we introduce vectors nf (as traceless Hermitian matrices, which can be naturally identified with vectors in R3 ) defined by: nf = Uf HUf−1

,

(2.2.11)

 0 i −i 0

.

(2.2.12)

where

 H=

For each pair {f, e} with e ⊂ f , we define vectors Bf e (also as traceless matrices): Bf e = je (2Uf Oφf e |1/2ih1/2|Oφ−1 U −1 − I) fe f    1 0 = Uf Oφf e je Oφ−1 U −1 . fe f 0 −1

(2.2.13)

Note that the length of Bf e is equal to je . We can already deduce that 0 ,

(2.2.49)

where we regard nf and Bf ei as vectors using the natural identification of hermitian matrices with R3 (tracial scalar product). Condition (2.2.49) fixes the sign of nf and also completes the geometric interpretation of the points of stationary phase.

2.2.7 Interpretation of planar (spherical) spin-networks as polyhedra In the last section we obtained an interpretation of the stationary points in terms of a set of vectors Bf e satisfying closure conditions for every face f X

Bf e = 0 .

(2.2.50)

e

However, these conditions do not specify a unique reconstruction of the according surface dual to the spin network. In fact, already each triangle allows for two different configurations of Bf e vectors. Therefore, we will here describe a method to reconstruct the surface from Bf e vectors for the spherical case: Let us draw the graph on the sphere (on the plane) as described in appendix 2.A.1. From the possible ways of drawing it, which in the case of 2-edge irreducible spin networks is in one-to-one correspondence with the orientation of the spin network, we have to choose one. In the case of 2-edge irreducible graphs the polyhedra obtained from different choices only differ by orientation. In addition to nodes and edges, there is also a natural notion of two-cells. The latter are defined as areas bounded by loops of edges. We are mainly interested in the dual picture that in this case is a triangulation of the sphere. Thus there is a unique identification of the vertices in the dual picture. A cyclic ordering of the edges for each f is inherited from the orientation of the sphere. In the following, we will construct an immersion (not an embedding) of this triangulation of the sphere into R3 , such that every edge e is given by Bt(e)e (with the right orientation). Let us choose one vertex v0 . Every other vertex v 0 can be connected to v0 by a path v0 , e0 , v1 , e1 , . . . , v 0 .

(2.2.51)

Every edge ei in the sequence belongs to two faces. Exactly one of these faces is such that vi , vi+1 are the consecutive vertices w.r.t the cyclic order of the face. We denote this face by fi (see figure 2.3). We introduce the vector X v˜0 = Bfi ei . (2.2.52) i

28

2.2 Modified coherent states, spin-network evaluations and symmetries One can prove that this vector does not depend on the chosen path. To see this, let us consider a basic move that consists of replacing vi , ei , vi+1 by vi , e, v, e0 , vi+1 where all three vertices belong to the same face f . Using the property X Bf e = 0 (2.2.53) e⊂f

and the proper orientation, one can show that the vector v˜0 is invariant with respect to this move. In fact any two paths can be transformed into one another by a sequence of these basic moves (or their inverses) due to the fact that the graph is spherical. A different choice of v0 gives a translated surface. It is straightforward to check that

v2

f1

v1

e1

e0 v0

Figure 2.3: Reconstruction of the surface

v˜b − v˜a = Bf e

,

(2.2.54)

where va and vb are vertices joint by the edge e and f is the face such that (va , vb ) is the pair of consecutive vertices in the cyclic order of f . Let us notice that from three vectors Bf e satisfying the closure condition one can form a triangle in two ways (see figure 2.4), but only that one depicted on the left appears in the reconstruction discussed here. Moreover the direction of the normal to the face coincides with the orientation

Bfe Bfe

Bfe

1

Bfe

2

Bfe

2

3

Bfe

1

3

Figure 2.4: Two possible triangles formed by Bf e satisfying closure condition.

inherited by the face from the cyclic order of its edges. For non-planar graphs, in general, we can only reconstruct the universal cover of the surface. Before we continue with the stationary point analysis for the angles φf e in the next section, let us briefly summarize the results of section 2.2: We have introduced a class of modified coherent states for irreducible representations of SU (2), which contain an additional smearing parameter,

29

2 Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions and presented how to construct invariants from them. From the contraction of these invariants (according to the spin network) an effective action has been derived, whose points of stationary phase allow for the same geometrical interpretation as the standard coherent states [A19, A20]. The amount of stationary points is significantly increased by the smearing parameters, yet they are all related by symmetry transformations of the action; a certain set of them can be suppressed by the prescribed modifiers. Eventually, we have depicted a way to reconstruct a triangulation from planar spin networks.

2.3 Variable transformation and final form of the integral In this section we focus on the stationary point analysis with respect to the angles φf e , which is the key modification in comparison to previously used coherent state approaches, see also section 2.2.1. This analysis allows us to obtain an effective action for Se associated to the edge e in terms of a phase, which we will identify as the angle between the normals of the faces sharing the edge e. Furthermore we are able to expand the effective action for Se in orders of 1j and initiate the discussion of next-to-leading order contributions.

2.3.1 Partial integration over φ and the new action Suppose that we have a non-degenerate configuration, i.e. ∀e ns(e) · nt(e) 6= ±1

.

(2.3.1)

Then the partial stationary point analysis with respect to all φf e can be performed. Its result will be the sum over the contribution from all stationary points with respect to φf e for a given configuration of Bf e vectors, but for fixed Uf (so also fixed nf ). Stationary points for Se In this section we will explain the contribution to the integral from the stationary point of the action Se with respect to φs(e)e , φt(e)e . The fs(e) , ft(e) terms can be ignored, since they are equal to 1 around the stationary point. We can separately consider terms corresponding to each edge Z
2j 1 dφs(e)e dφt(e)e  Us(e) Os(e)e |1/2i , Ut(e) Ot(e)e |1/2i e , (2.3.2) 2 4π and perform the stationary point analysis that gives the asymptotic result of the integration over φs(e)e , φt(e)e . The stationary point with respect to φt(e)e and φs(e)e is given by the conditions Us(e) Os(e)e |1/2i ⊥ Ut(e) Ot(e)e |1/2i ,

(2.3.3)

which is equivalent to 

−1 −1 U = Os(e)e Us(e) Ut(e) Ot(e)e



1 0   −iθ˜ 0 −1 0 −1 s˜ = (−1) e 1 0

,

(2.3.4)


where θ˜ ∈ − π2 , π2 and s˜ ∈ {0, 1} are uniquely determined by this equation. In section 2.3.1 we will show that 2θ˜ can be interpreted as the angle enclosed by the normal vectors ns(e) and nt(e) (w.r.t. the axis Bt(e)e ). Hence, Se on the stationary point is of the following form: Se = 2je ln (· · · ) = 2je θ˜ + i 2je π˜ s .

30

(2.3.5)

2.3 Variable transformation and final form of the integral As already discussed in section 2.2.5, each stationary point is characterized by the existence of Bt(e)e = −Bs(e)e orthogonal to both ns(e) and nt(e) (see also stationary point conditions in section 2.2.2). There exist two such configurations that differ by a sign of Bs(e)e . For every configuration one has 4 stationary points that can be obtained from one another by −os(e) - and −ot(e) -transformations. In case je is an integer the contributions from the two stationary points are equal, see also section 2.2.3. Contributions from Bf e configurations with opposite signs are related by complex conjugation. Geometric interpretation of the angle θ˜ ˜ Here we will provide The missing piece of the description above is the exact value of the angle θ. a geometric interpretation of this angle and its relation to the angle between faces. Let us recall: Bs(e)e

  1 0 −1 −1 = je Us(e) Os(e)e Os(e)e Us(e) 0 −1

(2.3.6)

−1 −1 ns(e) = Us(e) Os(e)e HOs(e)e Us(e)

(2.3.7)

Bt(e)e

nt(e) =

−1 −1 Ut(e) Ot(e)e HOt(e)e Ut(e)

iθ˜ |B

=e

t(e)e |

Bt(e)e

−iθ˜ |B

ns(e) e

t(e)e |

(2.3.8)

The angle 2θ˜ is the angle by which one needs to rotate ns(e) around the axis Bt(e)e to obtain nt(e) . We will denote this SO(3) angle by ˜ θ = 2θ,

θ ∈ (−π, π) .

(2.3.9)

This remaining ambiguity of the sign factor s˜ will be resolved in appendix 2.A.

2.3.2 Partial integration over φ We introduce new variables φ1 = φs(e) − φ0s(e) ,

φ2 = φt(e) − φ0t(e) ,

(2.3.10)

where φ0s(e) and φ0t(e) denote the stationary points. Then using (2.3.4), we can write the action as: 1 4π 2

Z

 2j ˜ ˜ d φ1 d φ2 (−1)2je s˜ eiθ cos φ1 cos φ2 + e−iθ sin φ1 sin φ2

,

(2.3.11)

where we integrate over φi . By splitting the terms in the bracket in real and imaginary part, we obtain: cos θ˜ (cos φ1 cos φ2 + sin φ1 sin φ2 ) +i sin θ˜ (cos φ1 cos φ2 − sin φ1 sin φ2 ) {z } | {z } | cos(φ1 −φ2 )

.

(2.3.12)

cos(φ1 +φ2 )

We define new variables α := φ1 − φ2 ,

β := φ1 + φ2

,

(2.3.13)

and the Jacobian for this transformation is given by: ∂α∂β 1 −1 ∂φ1 ∂φ2 = 1 1 = 2 .

(2.3.14)

31

2 Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions Hence equation (2.3.11) becomes: Z  2je 1 2je s˜ ˜ cos α + i sin θ˜ cos β = dαdβ (−1) cos θ 8π 2      Z     1 = 2 dαdβ (−1)2je s˜ exp 2je ln cos θ˜ cos α + i sin θ˜ cos β  8π   {z } | 

,

(2.3.15)

=:Se0

where Se = Se0 + i 2je π˜ s. Expansion around stationary points Given the definitions from the previous section, we compute the expansion of the following expression: Z 1 0 d α d β (−1)2j˜s eSe . (2.3.16) 2 8π The stationary point is given by α = β = 0, which corresponds to φi = 0, i.e. φs(e) = φ0s(e) , φt(e) = φ0t(e) . In this point the action associated to the edge e becomes:   ˜ Se0 = 2je ln eiθe = i2je θ˜e (2.3.17) In order to compute the first order contribution, one has to consider the matrix of second derivatives (evaluated on the point of stationary phase): ∂ 2 Se0 cos θ˜ cos α = − 2j e ∂α2 cos θ˜ cos α + i sin θ˜ cos β ∂ 2 Se0 ∂ 2 Se0 =0 = , ∂α∂β ∂β∂α ∂ 2 Se0 sin θ˜ cos β = − 2ij e ∂β 2 cos θ˜ cos α + i sin θ˜ cos β

,

(2.3.18) (2.3.19) .

(2.3.20)

Around the stationary point the action can be expanded (up to second order in the variables α, β): !  ˜ −iθ˜
−2je cos θe 1 0 α 0 α β Se = i2je θ˜ + + ··· . (2.3.21) ˜ −iθ˜ β 2 0 −2ije sin θe In order to correctly perform the stationary phase approximation, it is indispensable to state the
right branch of the square root, here for θ˜ ∈ − π2 , π2 : q p ˜ −iθ˜ = | cos θ|e ˜ −i 12 θ˜ cos θe
 −i π q p (2.3.22) e 4 θ˜ ∈ − π2 ,
0 ˜ −i 12 θ˜ −i θ ˜ ˜ i sin θe = | sin θ|e . i π4 π e θ˜ ∈ 0, 2 Let us notice that

 π π sgn sin θ = sgn sin θ˜ for θ˜ ∈ − , . 2 2 Hence, the leading order contribution from the stationary point is:    1 1 2π(−1)2je s˜ i2θ˜(j+ 1 )−i π sign(sin 2θ) ˜ 2 4 r e 1+O 2 8π j 2j 2 sin 2θ˜ In the next section we will show an improvement of this result.

32

(2.3.23)

.

(2.3.24)

2.3 Variable transformation and final form of the integral The total expansion of the edge integral Let us introduce a number (see appendix 2.H for a motivation of its origin) Cj =

1 Γ(2j + 1) 4j Γ(j + 1)2

.

(2.3.25)

 √  C We can multiply (2.3.24) by Cjj = 1 and use the expansion C1j = πj 1 + O( 1j ) derived in appendix 2.D.2 to write the result as    (−1)se 1 i(θ(je + 12 )− π4 sign(sin θ)) Cje p e 1+O . (2.3.26) je 4 2πje |sin θ| By se we denoted the sign factor   0 je integer, 0 je half-integer and s˜ = 0 se =  1 je half-integer and s˜ = 1 .

(2.3.27)

We will determine the sign se in appendix 2.A. We introduce new ‘length’ parameters le := je + se

and using the fact that √(−1) 4

2πle |sin θ|

= √(−1) 4

se

2πje |sin θ|

1 2

,

(2.3.28)


1 + O(j −1 ) we can express (2.3.26) in terms of le .

Before we move on, we would like to present a first glimpse at the next-to-leading order contribution: As it will be shown in section 2.5.2 by application of the stationary point analysis (2.3.26) and the recursion relation (2.5.29), the contribution (including next-to-leading order (NLO)) from the integral of eSe over φs(e)e , φt(e)e is given by    1 (−1)se i(le θ− π4 sign(sin θ)− 8l1 cot θ) e p Cje e 1+O 2 l 4 2πle |sin θ| e

,

(2.3.29)

where θ ∈ (−π, π) is the angle by which one has to rotate ns(e) around Bt(e)e to obtain nt(e) .

2.3.3 New form of the action In the previous sections we have computed the contribution of one point of stationary phase with respect to the angles φf e . From section 2.3.2 we can also conclude that having one stationary point ˜ that keep Uf fixed. These are all others are obtained by application of transformations from G given by compositions of (−uf )of (π), −of e ∀f . (2.3.30) However, only the orbit generated by the group of (−uf )of (π) from a non-trivial stationary point contributes, since all other stationary points are suppressed by the modifiers ff . Therefore it is sufficient to compute the number of these stationary points. The group generated by (−uf )of (π) |F | is equal to Z2 and acts freely on the stationary points; the countability of the orbit is thus 2|F | . Around the stationary orbit, the integral is hence of the form: Z Y 1 Y i(je + 21 )θe − cot θe 1 −i π4 sgn sin θe s |F | 8(je + 1 2) e e , (−1) 2 dUf (−1)se Cje q
4 2π je + 21 | sin θe | e (2.3.31)

33

2 Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions where θe is the angle between ns(e) and nt(e) with the sign determined by left hand rule with respect to Bt(e)e . In Q the neighbourhood of the stationary point this definition is meaningful. The value of the product e (−1)se is discussed in appendix 2.A.2. We use new ‘length’ parameters introduced in section 2.3.2 1 le := je + (2.3.32) 2 and perform a change of variables Uf → n f , (2.3.33) which is worked out in appendix 2.B.1. The correct integral measure is given by: µ=

1 δ(|n|2 − 1) dn1 dn2 dn3 2π

.

(2.3.34)

Thus, we can write the integral (integrating out of and −uf gauges) as: P Q π P (−1)s (−1) e se e−i 4 e sgne e Cje 5

1

2 2 |E| π |F |+ 2 |E| Z Y f ∈F

δ(|nf |2 − 1) d3 nf pQ

P le 1 ei e (le θe +S1 (θe )) e le | sin θe |

(2.3.35) .

where from 2.3.2 we know

1 cot θe + . . . . (2.3.36) 8le The only present symmetry that has to be discussed is a u-symmetry, which is implemented by SO(3) rotations: nf → unf u−1 (2.3.37) S1le (θe ) = −

If the configurations of the vectors Bf e is rigid, i.e. the only deformations of the configuration of the edges with given lengths are rotations, then the stationary u-orbit is isolated, i.e. there exists a neighbourhood of the orbit that does not intersect any other orbit. c transformation as parity transformation Furthermore, we would like to point out that given one orbit of stationary phase, we can always construct a different one via parity transformation of the Bf e vectors (see also section 2.2.5 about c transformations). After integrating out gauges these two points are related by n0f = nf

,

Bf0 e = −Bf e

,

(2.3.38)

so also the angles are related by θe0 = −θe (nf are preserved as pseudovectors). Finally, we see that the asymptotic contribution from the parity related stationary orbits is just the complex conjugate of the original one, such that the complete expansion is real. In order to provide the correct expression of the action before performing the remaining stationary point analysis, it is necessary to compute the normalization of the intertwiners, the so-called ‘Theta’ graph.

2.3.4 Normalization - ‘Theta’ graph We need to compute the self-contraction of the invariants Cf using the (in this case) symmetric bilinear form  (as a generalization of the anti-symmetric form  of spin 1/2 to arbitrary representations). Its special properties allow us to relate the  product ((·, ·)) to the scalar product on SU (2):     0 1 (Cf , Cf ) = Cf , Cf = hCf , Cf i , (2.3.39) −1 0

34

2.3 Variable transformation and final form of the integral since Cf is real and SU (2) invariant. The integral of the contraction of the intertwiner with itself is given by: Z Y d φi1 Y d φi2 dU1 dU2 2π 2π SU (2)2 ×S16 i i (2.3.40) 2ji Y −1 −1 1 1 f1 ({φi1 })f2 ({φi2 }) /2|, Oφi1 U1 U2 Oφi2 | /2 . i

Its stationary point conditions are: • Bi1 = −Bi2 . P P • i Bi1 = i Bi2 = 0 . As the ‘Theta’ graph itself is an evaluation of a spin network its effective action have the same transformations on the action as described in 2.2.3. The u symmetry can be ruled out just by dropping the integration over U1 . Then one is left with the group G generated by the transformations rf 1 , rf 2 , −u2 , of 1 , of 2 , −of 1,i , −of 2,i

.

(2.3.41)

On the stationary H orbits, i.e. the normal subgroup of G generated by {of , −uf }, these transformations act as the group K = G/H, which gives Z32 × Z32 . This group acts freely on the stationary H orbits and as before the modifiers suppress all but one P of the H orbits. If we take f1 = f2 = 1 and restrict ourselves to the case where i ji is even (all ji integer) then the action is invariant with respect to all transformations, thus every stationary orbit P contribute the same 216 of the overall result. In the case when je is not even, or some je are not integer, this choice leads to a vanishing invariant. The computation of the full expansion of the theta graph in the even case also gives an expansion on the stationary orbit in the presence of fi . This is briefly discussed in the next section. P

j even P We will derive the complete expansion for j even. We need to compute Y j1 j2 j3 2 Cji (C000 )

Theta graph for integer spins and

(2.3.42)

i

where Cji (see also appendix 2.D.2) is the normalization of the |0i vector. j1 j2 j3 2 In appendix 2.D.3 we show (following [A40]) that the theta graph (C000 ) is equal to    1 1 1+O 2 , (2.3.43) 2πS l where S is the area of the triangle with edges ji + 12 .

2.3.5 Final formula Let us state the final formula normalized by the square roots of the ‘Theta’ diagrams. Those are equal to: sQ    1 e⊂f Cje sf −7/2 (−1) 2 , (2.3.44) 1+O 2 πSf l where sf is a sign factor necessary to be consistent with [A38, A39] that will be derived in 2.A.2.

35

2 Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions To summarize the various calculations of this chapter, the contraction of normalized intertwiners has the following asymptotic expansion after the stationary phase approximation for the angles φf e has been performed and the asymptotic expansion from (2.3.44) has been inserted: P P Q 1/2 π P (−1)s+ f sf + e se e−i 4 e sgne f ∈F Sf Q 5 7 1 1 1/2 2 2 |E|− 2 |F | π 2 |F |+ 2 |E| e∈E le (2.3.45)  Z Y P  1 i e le θe − 8l1 cot θe 2 3 e δ(|nf | − 1) d nf pQ e . | sin θ | e e f ∈F P P As it will be shown in appendix 2.A, s + f sf + e se = 0 mod 2 and thus the term P

(−1)s+

f

P sf + e se

(2.3.46)

in the integral can be omitted. This is the contribution up to next-to-leading order. It is straightforward to generalize it to higher order due to the complete expansion of the edge amplitude (section 2.5.2) and the expansion of ‘Theta’ diagrams (appendix 2.D.3). In the next section we will focus our attention on the specific example of the 6j symbol. After another variable transformation to the set of exterior dihedral angles of the tetrahedron has been performed, we obtain the action of flat first order Regge Calculus , i.e. Regge Calculus in which both edge lengths and dihedral angles are considered as independent variables. The stationary point conditions (with respect to the dihedral angles) will reduce the action to ordinary Regge calculus, such that the geometry is entirely described by the set of edge lengths, where angles on the stationary point agree with the angles given for a tetrahedron built from the lengths. We will perform the stationary point analysis, in particular compute the determinant of the Hessian matrix, and obtain the correct asymptotic expression for the SU (2) 6j symbol [A5].

2.4 Analysis of 6j symbol and first order Regge Calculus In this section, we will perform the remaining integrations via stationary phase approximation starting from (2.3.45) in the case of the 6j symbol. As we are restricting the discussion to a specific spin network, we introduce the following notations: This spin network consists of 4 faces f , which we will simply count by i ∈ {1, . . . , 4}, and 6 edges e, which we will denote by ij, i < j, i.e. the faces sharing it. On the stationary point with respect to {φf e }, we have two configurations of Bf e , which we will label accordingly as Bij and similarly θij using the convention that θij is the angle at the edge lij . In [A19] it has been shown that the 6j symbol can be interpreted as a tetrahedron on the points of stationary phase (for non-degenerate configurations). In section 2.2.2 we have shown that our approach gives the same interpretation. Hence, we can assume that for one stationary point, the normals to the faces ni of the tetrahedron are outward pointing and the Bij vectors are oriented 0 = −B , the angles are such that θij ∈ (0, π). For the second stationary orbit, described by Bij ij negative, hence this contributes the complex conjugate. In order to perform the remaining stationary point analysis, it is necessary to perform another variable transformations from normals of faces ni to angles between these normals θij followed by integrating out gauge degrees of freedom corresponding to u transformations: ni → θij

.

(2.4.1)

This transformation is performed in appendix 2.B.2 in great detail, and we obtain the following relation: Y Y Y ˜ , d3 ni δ(|ni |2 − 1) → dθij | sin θij |δ(det G) (2.4.2) i

36

ij

ij

2.4 Analysis of 6j symbol and first order Regge Calculus ˜ denotes the angle Gram matrix (for exterior dihedral angles) of a tetrahedron with comwhere G ponents Gij = cos(θij ), with θii = 0. Using (2.4.2) and simplifying (2.3.45) for the case of the 6j symbol, we obtain in the neighbourhood of the stationary point:  6 Q P 1/2 Z Y Y i i 3,” Theor.Math.Phys. 131 (2002) 765–774, arXiv:math/0211165 [math-gt]. [C67] V. Bonzom, “From lattice BF gauge theory to area-angle Regge calculus,” Class. Quant. Grav. 26 (2009) 155020, arXiv:0903.0267 [gr-qc]. [C68] V. Bonzom, “Spin foam models for quantum gravity from lattice path integrals,” Phys. Rev. D 80 (2009) 064028, arXiv:0905.1501 [gr-qc]. [C69] V. Bonzom, “Spin foam models and the Wheeler-DeWitt equation for the quantum 4-simplex,” Phys. Rev. D 84 (2011) 024009, arXiv:1101.1615 [gr-qc]. [C70] V. Bonzom and A. Laddha, “Lessons from toy-models for the dynamics of loop quantum gravity,” SIGMA 8 (2012) 009, arXiv:1110.2157 [gr-qc]. [C71] A. Baratin and L. Freidel, “Hidden Quantum Gravity in 4-D Feynman diagrams: Emergence of spin foams,” Class.Quant.Grav. 24 (2007) 2027–2060, arXiv:hep-th/0611042 [hep-th]. [C72] D. Oriti, “The Feynman propagator for quantum gravity: Spin foams, proper time, orientation, causality and timeless-ordering,” Braz.J.Phys. 35 (2005) 481–488, arXiv:gr-qc/0412035 [gr-qc]. [C73] C. Rovelli private communication . [C74] H. W. Hamber, “Phases of simplicial quantum gravity in four-dimensions: Estimates for the critical exponents,” Nucl.Phys. B400 (1993) 347–389.

132

References [C75] B. Bahr, B. Dittrich, and J. P. Ryan, “Spin foam models with finite groups,” J. Grav. 2013 (2013) 549824, arXiv:1103.6264 [gr-qc]. [C76] B. Dittrich, F. C. Eckert, and M. Mart´ın-Benito, “Coarse graining methods for spin net and spin foam models,” New J. Phys. 14 (2012) 035008, arXiv:1109.4927 [gr-qc].

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5 Discretization independence implies non–locality in 4D discrete quantum gravity Bianca Dittrich, Wojciech Kami´ nski and Sebastian Steinhaus Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5

published in: Class. Quant. Grav. 31 (2014) 245009 , [arXiv:1404.5288 [gr-qc]]. Copyright 2014 IOP Publishing Ltd. http://dx.doi.org/10.1088/0264-9381/31/24/245009 Abstract The 4D Regge action is invariant under 5–1 and 4–2 Pachner moves, which define a subset of (local) changes of the triangulation. Given this fact one might hope to find a local path integral measure that makes the quantum theory invariant under these moves and hence makes the theory partially triangulation invariant. We show that such a local invariant path integral measure does not exist for the 4D linearized Regge theory. To this end we uncover an interesting geometric interpretation for the Hessian of the 4D Regge action. This geometric interpretation will allow us to prove that the determinant of the Hessian of the 4D Regge action does not factorize over 4–simplices or subsimplices. It furthermore allows to determine configurations where this Hessian vanishes, which only appears to be the case in degenerate backgrounds or if one allows for different orientations of the simplices. We suggest a non–local measure factor that absorbs the non–local part of the determinant of the Hessian under 5–1 moves as well as a local measure factor that is preserved for very special configurations.

135

5 Discretization independence implies non–locality in 4D discrete quantum gravity

5.1 Introduction Many quantum gravity approaches rely on a path integral construction as their foundation, for example spin foam models [D1, D2], group and tensorial field theory [D3, D4], (causal) dynamical triangulations [D5] or quantum Regge calculus [D6–D9]. These approaches have the same goal, namely to provide a way to compute (and give meaning to) the gravitational path integral, i.e. the sum over all histories between two 3–dimensional boundary geometries, where each history is a 4–geometry describing a possible transition weighted by the (exponential of the) Einstein–Hilbert action. An essential part in this path integral is the measure over the space of geometries, i.e. the space of all metrics modulo diffeomorphisms. To propose a well–defined path integral, one generically has to introduce a regulator to truncate the degrees of freedom of the theory. In gravitational theories mentioned above this is achieved by discretizing the theory, e.g. on a triangulation. However, the introduction of discretizations comes with a caveat: In general, a discretization of the classical theory cannot be chosen uniquely, if the only requirement is that this discretization leads to the correct continuum action. Whereas some agreement has been reached on the Regge action as a preferred discrete action [D10], at least for the theory without cosmological constant, the debate on the measure in Regge calculus [D11–D14] and spin foams [D15, D16] is not settled. Even more troubling in the context of gravity, discretizations generically break diffeomorphism symmetry [D17–D19], which is deeply intertwined with the dynamics of general relativity. Furthermore, this might induce an unphysical dependency of this theory on the choice of the discretization. Different approaches to quantum gravity differ in how they deal with these problems, e.g. in Regge calculus one considers only one triangulation with varying edge lengths, whereas in causal dynamical triangulations one keeps equilateral simplices and sums over all triangulations. Group field theories additionally sum over all topologies. Which of these schemes leads to a sensible theory of quantum gravity cannot be determined a priori. These intricacies are deeply rooted in the fact that diffeomorphism symmetry is broken by the discretisation and that the relation between discrete and continuous gravity is still hardly understood. This particularly affects the choice of measure in quantum gravity theories, which is crucially important for the dynamics in the continuum limit, since it also resembles a choice of the measure on the space of geometries. For spin foam models, arguments that link diffeomorphism symmetry, choice of (anomaly free) measure and divergence structure due to having non–compact gauge orbits from the diffeomorphism group have been made in [D15, D20]. This led also to the suggestion to choose a measure which has a certain (weak) notion of discretization independence, e.g. such that the amplitudes become independent under ‘trivial’ edge and face subdivisions [D16, D21–D23]. As mentioned the choice of measure heavily influences the divergence structure of the models [D23–D25]. Thus one can adjust the measure to obtain the divergence structure that fits the divergences one expects from diffeomorphism symmetry. This of course does not fully guarantee diffeomorphism symmetry or triangulation independence, as divergences might also arise due to other reasons. In Regge calculus, several measures have been proposed: in [D11] Hamber and Williams propose a discretization of the formal continuum path integral, with a local discretization of the (DeWitt) measure [D11, D12], conflicting with the proposal by Menotti and Peirano [D13], who mod out a subgroup of the continuum diffeomorphisms resulting in a highly non–local measure. A different discretization, also leading to a non–local measure due to discretizing first the DeWitt super metric [D26] and then forming the determinant, was proposed in [D14]. In this work we will pick up the suggestion in [D27], to choose a measure that, at least for the linearized (Regge) theory, leads to as much discretization invariance as possible. These considerations require to actually integrate out degrees of freedom and thus take the dynamics into account. The requirement of discretisation independence seems to be at odds with interacting theories, which possess local / propagating degrees of freedom. This apparent contradiction can be resolved

136

5.1 Introduction by allowing a non–local action or non–local amplitudes for the quantum theory – which in fact are unavoidable if one coarse grains the theory. Non–local amplitudes are however difficult to deal with. We therefore ask in this paper the question, whether in the quantum theory we can retain as much symmetry as in the classical theory, with a choice of local measure. The classical 4D Regge action is known to be invariant under 5−1 moves and 4−2 moves, but not under 3−3 moves [D27]. Here the non–invariance under the 3 − 3 moves – in fact the only move involving bulk curvature for the solution – allows the local Regge action to nevertheless lead to a theory with propagating degrees of freedom. We therefore ask whether it is possible to have a local measure for linearized Regge calculus that leads to invariance under 5 − 1 and / or 4 − 2 moves. Such a measure would therefore reproduce the symmetry properties of the Regge action. We will however show that such a local path integral measure does not exist. Our requirement of (maximal) discretisation independence is motivated by the ‘perfect action / discretisation’ approach [D28, D29] that targets to construct a discretisation, which ‘perfectly’ encodes the continuum dynamics and has a discrete remnant of the continuum diffeomorphism symmetry. Examples of such ‘perfect discretisations’ are 3D Regge calculus with and without a cosmological constant [D30] and also 4D Regge calculus, if the boundary data impose a flat solution in the bulk. In these examples, the basic building blocks mimic the continuum dynamics, e.g. one takes constantly curved tetrahedra for 3D gravity with a cosmological constant. Such perfect discretisations can be constructed as the fixed point of a coarse graining scheme, see for instance [D30–D32]. Once such a discretisation is constructed, the predictions of the theory become independent of the fineness of the discretisation. On the one hand one can compute observables for the coarsest discretisation, while on the other hand one can straightforwardly define the continuum limit and return to a description with local degrees of freedom. Indeed, the examples that have been considered so far lead to the conjecture that diffeomorphism symmetry is equivalent to discretisation independence. For quantum mechanical systems (with time discretization), it has been proven in [D32] that diffeomorphism symmetry implies discretization invariance. Recently, the relationship between diffeomorphism symmetry and discretization independence has been strengthened in the principle of dynamical cylindrical consistency [D33] for time–evolving discrete systems [D34], see also [D35–D38]. This conjecture has been the main motivation of [D27]: the question has been, whether requiring triangulation independence, i.e. invariance under Pachner moves [D39, D40], can be used as a dynamical determination of the path integral measure in (linearized) Regge calculus. One would expect that this requirement would also single out a unique measure. In 3D this lead to a simple measure factor invariant under all Pachner moves, which is consistent with the asymptotics of the Ponzano–Regge (spin foam) model [D41–D46]. The 4D case, that is the expressions for the determinants of the Hessians of the action evaluated on the solution, turned out to be astonishingly similar to the 3D one, yet these expressions were modified by an overall factor, which appears to be non–factorising with respect to the (sub)simplices of the triangulation and resisted so far a geometric interpretation. More importantly, it has been conjectured in [D27] to be non–local and, thus, effectively hindering a construction of a local path integral invariant under a subset of Pachner moves in 4D [D27]. This factor will be the main focus of this paper, in which we will derive a geometric interpretation, namely as a criterion determining whether d + 2 vertices (in d dimensions) lie on a (d − 1)– sphere [D47] (see also [D48]). This interpretation allows us to prove that this intricate factor is non–factorising, i.e. it cannot be expressed as a simple product of amplitudes associated to the (sub)simplices of the triangulation. (We will refer to this property as ‘non–local’.) This paper is organized as follows: In section 5.2 we review the setup and main results of [D27]. Section 5.3 deals with the derivation of the new interpretation of the non–factorising factor, namely as a criterion determining whether six vertices lie on a 3–sphere, uncovering the factor’s non–local

137

5 Discretization independence implies non–locality in 4D discrete quantum gravity nature. To examine the general cases, we have to generalize the study of [D27] to more general orientations in section 5.4. We discuss different choices for the measure (see also appendix 5.B). The paper is concluded by a discussion of the result in section 5.5. In appendix 5.A we discuss some particular cases of the non–local factor.

5.2 Linearized Regge calculus and previous results In the work [D27] it has been examined whether one can define a triangulation invariant path integral measure for (linearized) length Regge calculus. Let us briefly recap the setup: Consider the Euclidean path integral for the Regge discretization of gravity given on a 3D or a 4D triangulation: Z Y dle µ(le )e−SR [le ] , (5.2.1) le |e⊂∂M

e

where le |e ⊂ ∂M denotes the fixed edge lengths on the boundary of the triangulated manifold M . SR [le ] is called the Regge action and is given by the following expression in d dimensions: X X (bulk) (bdry) SR [le ] := − Vh ωh − Vh ωh , (5.2.2) h⊂bulk

h⊂bdry

(bulk)

(bdry)

where Vh is the volume of the (d − 2)–simplex h, also called ‘hinge’, and ωh and ωh denote the deficit angle or the exterior boundary angle respectively, located at the hinge h in the d– dimensional simplex σ d . The deficit angles at a hinge are defined as a sum of the dihedral angles of the d–simplices sharing the hinge modulo 2π and relative orientation of the simplices. The definition for matching orientations of the simplices, also considered in [D27], is: X (d) (bulk) := 2π − θh , (5.2.3) ωh σ d ⊃h

(bdry)

ωh

:= kπ −

X

(d)

θh

,

(5.2.4)

σ d ⊃h

(d)

where θh is the d-dimensional dihedral angle at the hinge h in the d–simplex σ d . k depends on the number of pieces glued together at this boundary. In case only two pieces are put together k = 1. This action (5.2.2) has been linearized, i.e. expanded (up to quadratic order) around a flat (0) background solution, denoted by edge lengths le , that is a solution of the equations of motion (bulk) 1 ∂SR : ∂le = 0 with vanishing deficit angles ωh le = le(0) + λe

.

(5.2.5)

The integration is then performed over the perturbations λe , such that the Hessian matrix of 2S R , becomes the (inverse) ‘propagator’ of the theory. The motivating the Regge action, i.e. ∂l∂e ∂l 0 e

(0)

question of [D27] has been whether it is possible to define a measure factor µ(le ), as a function of the background edge lengths, that allows for a triangulation invariant Regge path integral. To examine triangulation independence, it is enough to only consider local changes of the triangulation, so–called Pachner moves [D39, D40]: a consecutive application of these transfers a triangulation of a manifold into any other possible triangulation of the same manifold. In [D27] exact formulas for the Hessian matrix in 3D and 4D have been derived in great detail from which one concludes a very specific form of measure factors. In the following, we will restrict ourselves to just recalling the main results: 1

In 3D, the solution to the Regge equations of motion (for positive orientation) always implies vanishing deficit angles, however in 4D this is only possible if the boundary data admit a flat solution.

138

5.2 Linearized Regge calculus and previous results The Hessian matrices one has to compute are of the following form. In 3D we have ∂ωe ∂ 2 SR =− 0 ∂le ∂le ∂le0

,

(5.2.6)

where the dihedral angles are associated to the edges. In 4D the situation is more complicated X ∂Ah ∂ωh X ∂ 2 Ah X ∂ 2 Ah (bdry) ∂ 2 SR =− − ωh − ω ∂le ∂le0 ∂le ∂le0 ∂le ∂le0 ∂le ∂le0 h h

h

,

(5.2.7)

h

yet only the first term survives, since we are considering a flat background solution (ωh = 0) and only local changes of the triangulation, which leave the boundary unchanged. Thus for both cases of 3D and 4D Regge calculus, the main task is to compute the first derivatives of the deficit angles. Although these first derivatives of the deficit angles can be computed explicitly [D49], it is much more effective to use techniques of [D50, D51], see also [D52]. This utilizes the flat background (for the linearized Regge action) and hence the fact that the simplicial complex is embeddable into Rd . One then considers small deviations in the edge lengths so that the complex remains embeddable, i.e. the deficit angles ωh are unchanged, δωh ≡ 0. This requirement automatically translates into a requirement on the variations of the dihedral angles δθh , which are part of the respective deficit angle. Then one computes the derivatives of the dihedral angles [D49] under the assumption that only two edge lengths are varied at the same time [D50, D51]; starting from the simplest case, one derives all other derivatives of the deficit angles by considering the relative change of edge lengths under infinitesimal deviations. Eventually, one finds in 3D: lij lkm V¯i V¯j Vk¯ Vm ∂ωij ∂ 2 SR ¯ Q =− = (−1)si +sj +sk +sm + bdry terms ∂lij ∂lkm ∂lkm 6 ¯ n Vn

,

(5.2.8)

where lij is the edge lengths between the vertices i and j, ωij is the deficit angle at the edge (ij) and V¯i is the volume of the tetrahedron obtained by removing the vertex i 2 . The signs si depend on the orientation and the considered Pachner move. We provide a different derivation of them in section 5.3. In this work we will neglect the boundary terms, since they are not relevant for the main argument of this paper, yet they are essential to show that the classical Regge action is invariant under Pachner moves [D27]. Indeed, the idea to construct a triangulation invariant measure factor, relies crucially on the invariance of the classical Regge action under Pachner moves. In 3D this is the case for all Pachner moves, such that we were able to derive an invariant measure factor, which is factorising, and hence local, with a straightforward geometrical interpretation: Q 1 √ e 12π le µ({l}) = Q √ . (5.2.9) Vτ τ To each edge e of the triangulation one associates the edge length le and a numerical factor of 1 (12π)− 2 , to each tetrahedron τ one associates the inverse (square root) of its volume Vτ . This measure factor is consistent, even up to the numerical factor3 , with the asymptotic expansion of the SU(2) 6j symbol [D41–D46], the amplitude associated to a tetrahedron in the Ponzano Regge model, which is triangulation invariant (and topological) as well. In 4D Regge calculus the situation is more complicated, as one might expect: First the Regge action itself is in general not invariant under the 3–3 Pachner move [D27]. It is a peculiar move, since both possible configurations only differ by the triangle shared by all three 4–simplices of this 2

This assignment is unique in subsets of the triangulation, which are subject to a Pachner move. These subsets consist of d + 2 vertices in d dimensions; by removing one vertex, d + 1 vertices remain, which span a d–simplex. √ 3 The association of the factor ( 12π)−1 is not unambiguous. It can either be assigned to edges or tetrahedra.

139

5 Discretization independence implies non–locality in 4D discrete quantum gravity configuration, no dynamical edge is involved. Hence the configurations are solely determined by the boundary data, which might be chosen such that the deficit angle on the bulk triangle does not vanish, i.e. the configuration is not flat. However as soon as this is the case, the Regge action is not invariant under this Pachner move any more. Thus the Regge action is not a suitable starting point to define an invariant measure under all 4D Pachner moves. Nevertheless the 4D Regge action is invariant under the 5–1 and 4–2 Pachner moves (and their inverses), so it has been examined whether one can define a measure factor that is at least invariant under these two local changes of the triangulation. Surprisingly, the (considered part of the) Hessian matrix is very similar to the 3D case: X ∂Aopk ∂ωopk lop lmn Vo¯Vp¯Vm V ∂ 2 SR Q ¯ n¯ + bdry terms =− = Dop (−1)so +sp +sm +sn ∂lop ∂lmn ∂lop ∂lmn 96 l l V¯

,

k6=o,p

(5.2.10) where the factor Dop is the only difference to the 3D case, besides the fact that the V¯i now denote volumes of 4–simplices. Note that for the 5–1 move in 4D (and similarly the 4–1 move in 3D), the Hessian matrix possesses four null eigenvectors, which correspond to a vertex translation invariance of the subdividing vertex. The divergent part of the integral is then identified as an integral over a 4D volume and is gauge fixed to 1. We also provide a brief derivation of (5.2.10) in section 5.4 for more general orientations. The emphasis of this paper lies on this factor Dop , which is given by the following expression: X
2 2 2 − lop Vk¯ . (5.2.11) + lpk Dop := (−1)sk lok k6=o,p

Since the Hessian matrix is symmetric, one concludes that Dop actually does not depend on the vertices o and p, such that it turns into an overall factor of the Hessian matrix. Thus we will only refer to it as D in the following. Ignoring D for the time being, one can construct an ‘almost’ triangulation invariant measure, very similar to the 3D case: Q √ 1 e 192π le µ({l}) = Q √ , (5.2.12) ∆ V∆ where V∆ now denotes the volume of the 4–simplex ∆. The numerical factors are again assigned to the edges4 . Despite the concise expression (5.2.11) of D, it both impedes the construction of a triangulation invariant measure and resists a nice geometric interpretation, since it is not obvious, whether it can be written as a product of amplitudes associated to (sub)simplices. Hence it has been conjectured in [D27] that D is non–local and cannot be written in a factorising way. The purpose of this paper is to provide a geometric interpretation for the factor D, namely it is a criterion that determines whether the 6 vertices, that make up the simplicial complex (in 4D) to which the Pachner move is applied, lie on a 3–sphere. We will use this to prove that the factor D generically cannot be accommodated by a local measure factor, in particular not by simple product (or quotient) of volumes of (sub)simplices.

5.3 A geometric interpretation for D In this section we will derive a geometric interpretation for the factor D mentioned above. Actually, the definition (5.2.11) of D is valid in any dimension d ≥ 3, such that we will derive its geometric interpretation for arbitrary dimensions5 . 4 5

As in 3D, the assignment of numerical factors is not unambiguous. Note that it is not clear whether D will also arise in higher dimensions in the framework of linearized Regge calculus. We can only confirm this for d = 4 [D27].

140

5.3 A geometric interpretation for D In the following we will discuss a simplicial complex in d dimensions, to which a Pachner move will be applied. Consider d + 2 vertices embedded in Rd , such that they form non–degenerate d–simplices. The geometry can be completely characterized by the set of the edge lengths {lij }, describing the Euclidean distances between the vertices. Given a set of vertices and the edge lengths between them, one can define the associated Cayley–Menger matrix C [D47,D50,D51,D53]. In the case we consider here, this is a (d + 3) × (d + 3)–dim. matrix given by:  0 1 1 1 2 2 1 0 l01 l02   2 2 1 l01 0 l12 C :=  . .. .. .. . . . . . 2 2 2 1 l0(d+1) l1(d+1) l2(d+1)

··· ··· ··· .. . ···

1



2  l0(d+1)   2 l1(d+1)  ..   .  0

.

(5.3.1)

In general, the determinant of the Cayley–Menger matrix, det C, associated to a d–simplex is proportional to the square of its d–volume. However, in the example at hand, the d + 2 vertices are embedded in Rd , such that they form a degenerate (d + 1)–simplex, hence det C = 0. Since we have required that the d–simplices are non–degenerate, C has exactly one null eigenvector, corresponding to changes of the edge lengths, such that the d + 2 vertices remain (embeddable) in Rd . To describe this null eigenvector, let us introduce some notation. By Cji we denote the submatrix of C obtained by deleting its i’th column and its j’th row with i, j ∈ {0, 1, 2, . . . , d + 2}. The determinant of the submatrix, i.e. the (i, j)th minor of C, is denoted i by Cj := det Cji . To simplify notation we simply call the diagonal minors |Ci |. In fact, since det C = 0, all off-diagonal minors can be expressed in terms of the diagonal ones [D54]: q i p Cj = |Ci | |Cj | .

(5.3.2)

Before we construct the null eigenvector of C, it is instructive to examine the minors C0 and Ci for i > 0 in more detail and discuss their geometric interpretation. The matrix C0 , obtained by deleting the ‘0’th column and row of C is particularly important in this paper. It is given by: 

0

(l01 )2 0 .. .

... ...

 (l01 )2  C0 =  ..  . ... (l0(d+1) )2 (l1(d+1) )2 . . .

 (l0(d+1) )2 (l1(d+1) )2    ..  .

.

(5.3.3)

0

It has been shown in [D47] (see also [D48]) that |C0 | has a very specific geometric meaning. In case |C0 | = 0 the d + 2 vertices lie on a (d − 1)–dimensional sphere, see fig. 5.1 for an example in d = 2. In a sense, this is a non–local statement, since it can only be deduced if the positions of all d + 2 vertices are known; it cannot be inferred p from just d + 1 vertices. From the construction of the null eigenvector, we will show that D ∼ |C0 |. Hence we argue that D is non–local in section 5.3.1. The second interesting minor we want to discuss is |Ci | for i ∈ {1, 2, . . . , d+1}. If one takes a look at the definition of C in (5.3.1) again, then one realizes that by removing the i’th row and column, one removes all edge lengths with the index i − 1. The remaining matrix is again a Cayley-Menger matrix, yet for a d–simplex. Hence |Ci | = (−1)d+1 2d (d!)2 Vi−1


2

.

(5.3.4)

141

5 Discretization independence implies non–locality in 4D discrete quantum gravity

Figure 5.1: The situation in 2D: Four vertices on a 1–sphere.

The single null eigenvector of C is given by6   p p p ~vc := (−1)s |C0 |, (−1)s0 |C1 |, . . . , (−1)sd+1 |Cd+2 |   p = (−1)s |C0 |, (−1)s0 2d/2 d! V¯0 , . . . , (−1)sd+1 2d/2 d! Vd+1

(5.3.5) .

(5.3.6)

The signs s and si are determined only up to an overall sign ambiguity. We will show below that we can choose si to be the same signs that appear also in (5.2.10) and (5.2.11). The action of the matrix C on ~vc gives the following relations: d+1 X (−1)si V¯i = 0 ,

(5.3.7)

i=0

∀j≥0

d+1 X p 2 V¯i = 0 . (−1)si 2d/2 d! lij (−1) |C0 | + s

(5.3.8)

i=0

The first condition fixes the relative signs si and is straightforward to interpret if one recalls the change of the triangulation under the Pachner move. Take a 1 − (d + 1) move for example, where the new vertex 0 is added inside to the d + 1 vertices forming the d–simplex. The relation shows that the volume before and after the Pachner move is the same: V¯0 =

d+1 X

V¯i ,

(5.3.9)

i=1

which fixes the signs s0 = 1 and si = 0 ∀ i ≥ 1, consistent with the results of [D27]. Similar relations also hold for the other Pachner moves. For different (relative) orientations (see section 5.4), one or more signs get flipped such that such a clear separation between volumes ‘before’ and ‘after’ the Pachner move is not possible any more. In fact, since we mainly discuss the 5–1 move in this paper, we fix s0 = 1, i.e. the orientation of the coarse simplex, because it is determined by the boundary data and is thus not affected by moving the vertex 0. This singles out the coarse simplex as a particular reference frame with respect to which the relative orientation of the other simplices is defined. At this point we can also determine the sign s using equation (5.3.8) for j = 0 s

(−1)

p

d/2

|C0 | = −2

d+1 d+1 X X si 2 d/2 2 d! (−1) li0 V¯i = −2 d! li0 V¯i < 0 . i=1

i=1

Thus we conclude s = 1. 6

The null eigenvector can be deduced from the expansion of det C with respect to different rows.

142

(5.3.10)

5.3 A geometric interpretation for D The relations (5.3.7) can be used to derive a relation between the non–local factor D and the criterion |C0 | determining whether (d + 2) vertices lie on a (d − 1)–sphere. Recall that for arbitrary i 6= j, D was defined as: X 2 2 2 (5.3.11) Dij = (−1)sk (lik + ljk − lij )Vk¯ . k6=i,j

Let us expand Dij as follows: X 2 2 + ljk )Vk¯ Dij = (−1)sk (lik |k

(5.3.7)


X 2 Vk¯ (−1)sk lij − (−1)sj lij V¯j + (−1)si lij V¯i − k6=i,j

{z

= −2(−1)s 2−d/2 (d!)

√ −1

= − 21−d/2 (d!)−1 (−1)s

} |

{z

|C0 |

2 =−lij

P

}

(5.3.7) sk ¯ = 0 k (−1) Vk

p |C0 | .

(5.3.12)

This proves the relation between the non–local measure D and the criterion |C0 |. p This automatically gives a new interpretation to the non–factorising factor D ∼ |C0 | appearing in the 4D Pachner moves. If it vanishes7 all six vertices of the 4–simplices involved in the Pachner move lie on a 3–sphere. To determine whether this is the case or not, one has to know the positions of all six vertices with respect to each other, it cannot be inferred from a subset. Thus it is already implied that the factor D has to be non–local, since its geometric meaning can only be deduced if the relative positions of all six vertices are known. We will use this fact in section 5.3.1 to show that D does not factorise. This geometric interpretation is even more pronounced if we express the factor D in affine coordinates. To this end we specialize to the (d + 1) − 1 move, in which we integrate out d edge lengths, that start from a subdividing vertex 0, which lies inside the final simplex. An efficient way to describe the coordinate of the subdividing vertex 0 with respect to the final simplex ¯0 is by using affine coordinates. The idea is to write the position vector ~x0 of the Pnew vertex as a weighted sum of the position vectors ~xi , i 6= 0, with weights αi . The condition i6=0 αi = 1 ensures that this prescription is well–defined. If one additionally requires that αi ≥ 0, ∀i 6= 0, then the new vertex is inside the final simplex. As soon as one of the αi is negative, the vertex 0 is located outside. Hence, the position vector ~x0 is given by X ~x0 = αi ~xi , (5.3.13) i6=0

thus

 ~xk − ~x0 = 

 X i6=0

αi  ~xk −

X

αi ~xi =

i6=0

X i6=0

αi (~xk − ~xi )

.

(5.3.14)

2 = (~ The (square of the) new edge lengths is given by l0k xk − ~x0 )2 : X 2 l0k = αi αj (~xk − ~xi ) · (~xk − ~xj ) i6=0,j6=0

  1 X αi αj (~xk − ~xi )2 + (~xk − ~xj )2 − ((~xk − ~xi ) − (~xk − ~xj ))2 2 i6=0,j6=0   X X X X X
1 2 2 2 2  2 2 + ljk − lij = αi lik αj  − = αi αj lik αi αj lij = αi lik − b2 2 0 Nnew variables, the first set of equations will give Nold relations for Nnew unknowns. Thus, if the equations are independent, they will give the solutions q 0 (q, p) but also (Nold −Nnew ) pre–constraints Ci (q, p), i = 1, . . . , (Nold −Nnew ), that is constraints on the initial phase space. Similarly we obtain post–constraints, if Nnew > Nold . (Constraints can also appear independently of this mechanism, that is Nnew = Nold does not guarantee that constraints will not appear.) As the pre– or post–constraints are defined via a generating function, they are first class. Thus the evolution equations (7.2.1) leave a number of configurations undetermined (’pre–and post gauge degrees of freedom), corresponding to the number of constraints that appear. The status of these gauge degrees of freedom might change under further evolution: constraints appearing in the future might lead to a gauge fixing. The pre–constraints have to be satisfied for an evolution move to take place. The post–constraints are automatically satisfied, after an evolution move has taken place. We should point out that the pre– and post–constraints include constraints which might arise due to gauge symmetries, including Hamiltonian and diffeomorphism constraints. For instance the 4 − 1 Pachner move in 4D leads to Hamiltonian and diffeomorphism constraints [F10, F53]. In this case the post–constraints exactly coincide with the Hamiltonian and diffeomorphism constraints, no new truly physical degree of freedom is added by such a refinement move. There are however also the 2−3 moves that add degrees of freedom and therefore lead to constraints, which do however not boundary (i.e. multiplying the state with the simplex amplitude and summing over the variables which are now bulk variables), we automatically obtain the amplitude for the evolved state. Hence one would expect that the semi–classical limit reproduces the equation of motion as obtained from the canonical transformation generated by the action associated to this simplex. 5 Some configuration variables are neither old or new, as these are represented in the hypersurface before and after the move. Such variables count as (non–dynamical) parameters in this move. Here we will only need this schematic discussion, for explicit discussion of all Pachner moves see [F10].

209

7 Time evolution as refining, coarse graining and entangling coincide with the Hamiltonian and diffeomorphism constraints. As noted further evolution might fix the gauge degrees of freedom implied by the Hamiltonian and diffeomorphism constraints. This is due to the breaking of diffeomorphism symmetry in discretization of 4D gravity [F25, F26] Post– and pre– constraints also appear for theories without any a priori gauge symmetries, such as a scalar field theory. We propose here that such constraints can be interpreted as describing the state of finer degrees of freedom. We will motivate this proposal with examples.

7.2.1 Example: evolution of a scalar field on an extending triangulation As a first example we consider a massless scalar field on a 2D (Euclidean) equilateral triangulation. The action associated to one triangle is given as 1X S∆ = (φs(e) − φt(e) )2 , (7.2.2) 4 e⊂∆

where s(e), t(e) denote the source and target vertex of an oriented edge respectively. We now consider a time evolution between two spatial periodically identified 1D triangulations, i.e. circles subdivided into edges. We assume that the earlier equal time hypersurface has N edges and the later one N 0 edges and we connect these two hypersurfaces by “one slice of triangles”, see figure 7.3. The triangulation can be described by an adjacency matrix Avv0 , where Avv0 = 1 if the vertex v at the earlier time is connected to the vertex v 0 at the later time, and Avv0 = 0 if this is not the case. The canonical time evolution map can be easily computed in this case. In particular the momenta πv0 0 at the later time step are given by ! X ∂S πv0 0 = (7.2.3) = Avv0 (φ0v0 − φv ) + φ0v0 − 12 φ0v0 +1 − 21 φ0v0 −1 , ∂φv v where S is the action associated to the interpolating triangulation obtained by summing the action 0 0 contributions (7.2.2) of the triangles. If Avv0 has right null vectors Rrv , i.e. such that Avv0 Rrv = 0, we obtain constraints by contracting (7.2.3) with these null vectors. These constraints are of the form X 0 Cr = Rrv πv0 0 + fr (φ0 ) (7.2.4) v0

with some specific functions fr of the fields φ0v0 at the later time step. Coming from a generating function the constraints are Abelian. They generate gauge transformations in the sense that the evolution step leaves indeed certain combinations of field values unspecified. Assuming an orthogP 0 v onal basis of the null vectors, these combinations are given as λr = v0 Rr ψv0 0 . For a refining evolution step we have a larger number N 0 of vertices at the later time than the number of vertices N at the earlier time, N 0 > N . In this case there are at least N 0 − N right null vectors Rr , with r = 1, . . . , N 0 − N . We want to argue that these gauge degrees of freedom correspond to the finer degrees of the field. Consider specifically a regular refinement as in figure 7.3, with N 0 = 2N . The corresponding adjacency matrix is given by Avv0

= δ2v−1,v0 + δ2v,v0 + δ2v+1,v0

.

(7.2.5)

The matrix can be ‘diagonalized’ by a Fourier transform. Let us formally introduce the notation |ki = |k 0 i =

210

PN

v=1 e

P2N

v 0 =1 e

−2πi kv N

|vi

0 0

v −2πi k2N

|v 0 i

N −1 1 X 2πi kv |vi = e N |ki N

,

,

|v 0 i =

k=0 2N −1 X

1 2N

k0 =0

k0 v 0

e2πi 2N |k 0 i

,

(7.2.6)

7.2 Time evolving phase spaces 1•0 2•0 3•0

4•0

• 1

5•0

• 2

6•0

7•0 8•0

• 3

1•0

• 4

Figure 7.3: The scalar field on a circle and its time evolution. The circle is drawn as an interval with periodic boundary conditions indicated by the dashed lines.

so that we can write hk|A|k 0 i, =

X vv 0

hk|viAvv0 hv 0 |k 0 i =



 k0 k0 1 + eπi N + e−πi N δ (N ) (k, k 0 )

      kπ kπ 0 δ(k, k ) + 1 − 2 cos δ(k + N, k 0 ) (7.2.7) . = 1 + 2 cos N N 0

We thus have N right null vectors Rkk (here k labels the null vectors) given by X k0

0 Rkk |k 0 i

 :=

 1 − 2 cos

kπ N



 |ki −

 1 + 2 cos

kπ N

 |k + N i

.

(7.2.8)

In general the coefficient in front of the higher momentum |k + N i is non–vanishing (it only vanishes for k = 2N/3). Thus we can say that (almost) all momenta πk0 0 associated to finer degrees of freedom, i.e. with momenta k 0 > N are determined by the constraints (7.2.4). The same holds if Pwe add a potential V to the scalar field, and discretize this in a local manner, i.e. as a term v∈∆ Ar(∆, v) V (φv ) added to the action (7.2.2) with Ar(∆, v) denoting some association of an area to the vertex–triangle pair.

The post–constraints signify in particular that no new information is added, the physical phase space cannot be enlarged during evolution.6 In fact, we can interpret this in the following way: given a state with a certain coarse graining, i.e. discretization scale, we can apply refining time evolution steps. This will lead to a state with the same coarse graining scale, however represented on a finer discretization. It is preferable that the finer degrees of freedom that are added during refining time evolution are in a vacuum state. In the case of a scalar field we have a notion of energy, thus the statement is that the refining time evolution should not increase the energy as defined by some energy functional on the two different discretizations. See also the discussion in [F75], which considers this issue however in a covariant quantization scheme. Whether this is actually the case will depend on the quality of the discretization. This is particularly true because the degrees of freedom that are added are typically defined on scales near the discretization scale. Typical discretizations will rather give unreliable results on this scale. A way out is to design discretizations, so that the added degrees of freedom are in fact in a vacuum state. 6

For a complete discussion on how these constraints propagate and a classification of the constraints, see [F11].

211

7 Time evolution as refining, coarse graining and entangling

7.2.2 Refinement as adding degrees of freedom in the vacuum state In the last section 7.2.1 we have used the Fourier transform to identify finer degrees of freedom (higher modes) and coarser degrees of freedom (lower modes). In fact in a free theory the Fourier modes decouple and allow us to assign an energy per mode. We can thus make the argument that refining time evolution should add degrees of freedom in a vacuum state and design a discretization for which this is the case. This will in general result in a non–local (in space) discretization – as can be already suspected if one uses the Fourier transform. The general idea is to match for a set of Fourier modes up to a cut–off K exactly the dynamics of the continuum, see also the related arguments in [F76, F77]. The construction is as follows: We consider the canonical data of a continuum scalar field on a 1D circle, representing the spatial hypersurface. We only consider fields that include Fourier modes ˜ 1 ) up to a cut-off K ∈ N on the spatial momentum component |k1 | ≤ K, so that the fields are φ(k superpositions of N = 2K + 1 modes. Such fields can therefore be parametrized one–to–one (in the general case) by the set of N values of the field at pre–specified positions along the circle. We thus have a mapping MK from the phase space describing continuum field configurations with a cut-off K to a phase space describing a discretized scalar field on N = 2K + 1 vertices. We have now to decide on an embedding map CK,K 0 from the phase space with cut–off K to a phase space with K 0 ≥ K. Once such a map is chosen we can construct the corresponding map DK,K 0 = MK 0 ◦ CK,K 0 ◦ (MK )−1 for the phase spaces describing the discretized fields. As an example, one can choose CK,K 0 such that in a mode expansion the coefficients of the additional modes are vanishing. This minimizes the energy of the additional modes for a free theory. For interacting theories one can choose more generally EK,K 0 such that the energy of the refined configurations (in the space of fields with a mode cut-off K 0 ) is minimized, keeping the coarser modes fixed. Note also that one can attempt to define an embedding CK,K 0 such that it includes some proper time evolution step T . However a time evolution will entangle the modes up to the cut-off K, with (possibly) all continuum degrees of freedom, that is the image of T applied to a phase space given by modes with cut–off K will in general include modes K 0 → ∞. We therefore face a problem in pulling back the evolved continuum configuration to a discrete one, as the evolved continuum configuration might be infinitely refined with respect to the embedding maps chosen. Thus one must find an (embedding) map of discrete configurations into continuous ones, where this is not the case. In this sense the choice of the (generalized) maps MK should be informed by the dynamics. We believe however that such examples are rare, see section 7.2.3 for one of these cases. Alternatively, one chooses a truncation of the image of T back to the phase space describing modes with a cut–off K 0 . This introduces an approximation to the continuum dynamics. However for sufficiently refined configurations, one would of course expect, that these errors do not affect sufficiently coarse grained observables. In general the discrete embedding maps DK,K 0 will be highly non–local, but this will be necessary to obtain a good approximation also for modes near the discretization scale. This construction will be very involved for an interacting theory, as it basically requires the solution of the dynamics. However it can serve as a guide line of what to expect from ‘good’ discretizations, that also involve a possible change of degrees of freedom. Thus even if one has a discretization that does not exactly mirror this behaviour (i.e. is not ‘perfect’) one can hope that via coarse graining one reaches an effective theory, that actually does so. This is the philosophy behind perfect discretization, which can be constructed as fixed points of renormalization flows [F49,F78–F80] or by pulling back continuum physics to the lattice (‘blocking from the continuum’) [F81]. A more abstract approach is to select as observables spectra of geometric operators [F82]. The construction described in this section allows a more explicit choice of the (post)–constraints, than in the discussion in section 7.2, where the post–constraints are determined by the chosen discretization of the action. Here the post–constraints are determined by the choice of vacuum

212

7.2 Time evolving phase spaces state, given by the minimization of an energy functional. Note that the constraints are second class. For instance for a free theory these are given by the vanishing of all higher modes in the fields and momenta. Thus in comparison with the discussion in section 7.2 one has gauge fixed the first class post–constraints appearing there with additional constraints. This is what one would however expect also from the quantization of a (free) scalar field: the vacuum functional in a given mode is given as a Gaussian of the field variable. Such a Gaussian can also be found by minimizing the ‘master constraint’ M (k1 ) = ω 2 (k)φ˜2 (k1 ) + π ˜ (k1 ), given as sum of (weighted) squares of the individual constraints [F83–F85]. For gravity the situation is less clear what kind of vacuum to expect. On the one hand (continuum) gravity constitutes a first class constrained system, so all physical states have to satisfy these constraints. Thus, physical states are squeezed states in the conjugated degrees of freedom describing the gauge choice and the constraints. In 4D we of course have additional physical degrees of freedom, however the characterization of a vacuum state (without a background and boundary) is an open issue. We will discuss in section 7.3.2 the Hartle Hawking no–boundary proposal [F86] for a vacuum state, that can be naturally implemented with a refining time evolution.

7.2.3 Massless scalar field in a 2D Lorentzian space time Here we will discuss an example of a perfect discretization with local embedding maps, namely the discretization of a massless scalar field in 2D Minkowskian space time. Note that this is the only such example of a non–topological theory that we are aware of, and that the locality of the embedding maps might actually change in the quantum theory. We will consider equal time hypersurfaces given by piecewise null lines, akin to characteristic evolution schemes [F87]. We will identify a given discrete configuration of field values with a continuous configuration by assuming the continuum field to be piecewise linear. Such a piecewise linear field can be parametrized by a discrete set of scalar field values at points where the derivative of continuum field is not continuous. One motivation for this example is to provide an interpretation for graph changing Hamiltonians appearing in loop quantum gravity [F45–F47, F88, F89] or for the parametrized scalar field [F90– F92]. This example will illustrate that indeed refining evolution splits into an embedding map and a proper evolution. The example is furthermore interesting as it introduces the concept of piecewise null hypersurfaces, that on ‘larger scales’ can be either put together to a spatial hypersurface, or alternatively to a null hypersurface. Thus problems involving a null boundary can be easily treated, with a natural specification of ‘boundary conditions’ at the null hypersurface (or null line). For a discussion of issues related to holography involving such discretizations, see [F93], which very much inspired the development of this example. Null surface formulations also attracted recent interest in (loop) gravity [F94–F97]. To be concrete we consider a 2D cylinder space time endowed with the Minkowski metric. We consider piecewise null ‘hyperlines’ that close around the cylinder. Thus we will have null edges connected via kinks. For every such kink we have to introduce a vertex ν. We allow furthermore vertices ν on the null edges themselves. We will associate scalar field values φν to these vertices ν. As we will show in the following, such a configuration of scalar fields φν specifies a piecewise linear solution to the continuum dynamics. Let us start with the set of continuum solutions to φ = 0, which are given by φ(u, v) = f (u) + g(v)

(7.2.9)

where (u = t + x, v = t − x) are light cone coordinates7 . We will consider functions of the form (7.2.9) with f and g piecewise linear (continuous) functions. Thus φ(u, v) will be smooth (even 7

Choosing x ∈ [0, 2π) we have

1 (u 2

− v) ∈ [0, 2π).

213

7 Time evolution as refining, coarse graining and entangling ν10 • ν1  •

• ν3 • ν2

• 0 ν2

Figure 7.4: The time evolution proceeds by moving the null edge (ν1 , ν2 ) to (ν10 , ν20 ).

linear) everywhere except at a set of null lines u = cI or v = c0J , where {cI , c0J } are a set of constants. Such a solution induces a scalar field configuration on any piecewise null line in the following way: As outlined above, we have a vertex at every kink of the piecewise null line. Additionally we introduce vertices for every null line u = cI or v = c0J that cuts our ‘equal time hypersurface’ transversally. The values of the scalar field at these vertices are now just given by the values of the solution (7.2.9) at the position of the vertices. This gives a configuration of scalar field values on a piecewise null hyperline. From this configuration we can re–construct the solution. We basically do the inverse of the above procedure: For every kink we draw two null lines u = cI and v = c0J emanating from this kink. Furthermore we draw from every vertex on a null edge a transversal null line. These null lines give the possible non–smooth behaviour of the solution. We can reconstruct the solution everywhere by linear extrapolation. A concrete way of constructing such a solution is given by a time evolution of the scalar field configuration on a piecewise null line, by pushing the null line forward in time. Note that the scalar field values on a given piecewise null line are sufficient to reconstruct the full space time solution. There are no additional momenta needed. Intuitively this can be imagined the following way: drawing a null zigzag line we obtain a set of initial fields at two consecutive time steps. Thus the fields themselves provide the momenta. To describe the time evolution consider a piecewise null line with a vertex ν1 at a kink. We wish to evolve this vertex by an amount  in say the direction of u, see figure 7.4. (Note that there is no absolute length attached to  as it is an affine parameter. The time evolution itself can only be characterized by the area of the rectangular diamond that will be glued to the hypersurface.) Let us assume (for simplicity) that the next vertex ν2 to the right of ν1 is also a kink. If we want to move the vertex ν1 and keep the hypersurface null we have to also move the vertex ν2 to a new vertex ν20 . Thus we move an entire null edge of the hyperline. Note however that although we move the vertex ν1 to a new vertex ν10 we actually have to keep the vertex ν1 as a vertex in our hypersurface. This is due to the possible non–smooth behaviour in the field that might still occur at ν1 . Thus the new hypersurface will have one additional vertex. In this way time evolution is necessarily8 refining. Thus we have to determine the values of the scalar field at the new vertices ν10 and ν20 . We construct the field φ(ν20 ) by linearly interpolating between φ(ν2 ) and the field φ(ν3 ) at the next vertex ν3 to the right of ν2 : φ(ν20 ) = φ(ν2 ) +

φ(ν3 ) − φ(ν2 ) (u(ν20 ) − u(ν2 )) . u(ν3 ) − u(ν2 )

(7.2.10)

(One might consider this step to be somewhat non–local.) Having constructed the field φ(ν20 ) we now know three fields at the vertices of the diamond formed by ν1 , ν10 , ν2 , ν20 . The field at the tip 8

One can also time evolve by gluing diamonds that fit into the zigzag null line. This would keep the number of vertices constant, but would also pre–define the size of the time evolution step.

214

7.2 Time evolving phase spaces ν0 ν 00

Figure 7.5: Time evolution can also proceed by gluing small diamonds to the hypersurface, which will however produce past directed null edges. Note that in this case φ(ν 00 ) will be constrained and determined by the fields at the other vertices on the ‘equal time’ hypersurface.

ν10 of this diamond is imposed by the form of the solution (7.2.9) to be φ(ν10 ) = φ(ν1 ) + φ(ν20 ) − φ(ν2 )

.

(7.2.11)

This description can be easily generalized to other situations. In the end time evolution proceeds by gluing rectangular diamonds to the null hypersurface. Allowing the side length of these diamonds to vary, we do not pick out a Lorentz frame, thus we have a Lorentz independent cut–off. Here we have a situation very similar to loop quantum gravity [F45–F47], or parametrized and polymerized scalar field theory [F90–F92], with a ‘graph changing’ Hamiltonian. One can choose  (or the area of the diamond if one generalizes the framework to allow also ’past directed’ null edges, see figure 7.5) arbitrarily small – there remains a discontinuous action of the time evolution, which is to produce new vertices. The time evolution map can be also split into two parts: one is a pure refining part, introducing the vertex ν20 and the associated field value (7.2.10). The second part is the ‘proper’ time evolution step, in which a diamond with vertices ν1 , ν10 , ν2 , ν20 is glued to the hypersurface. This part keeps the number of vertices constant, as the new vertex ν10 is compensated by the loss of the old vertex ν2 . The scheme we have described is a perfect discretization, that exactly mirrors the continuum solutions. We can also embed any discrete configuration into a refined configuration, where the new field values are given by linear interpolation as in (7.2.10). This allows to identify discrete and continuum configurations, which can be formalized into an inductive limit construction, which we will explain in section 7.3.1 for the quantum theory. See [F98, F99] for an alternative proposal to identify discrete and continuous phase space configurations, based on gauge fixing infinitely many degrees of freedom of the continuum theory.

7.2.4 Refinement moves in simplicial gravity: adding gauge and vacuum degrees of freedom The examples discussed here involved scalar field theories, for which a notion of energy and hence vacuum is available. In section 7.2.1 we proposed to use directly an energy functional to characterize the state of the additional degrees of freedom. For the massless scalar field on null lines we determine the values of additional ‘finer’ fields by demanding piecewise linearity of the field. In case of gravitational theories the notion of energy is less clear, in particular if one considers compact spatial slices.9 Time evolution itself is rather understood as a gauge transformation and energy is constrained to vanish (on space times with compact spatial slices). Indeed as mentioned in section 7.2, part of the degrees of freedom added in a refining time evolution will be gauge (or 9

However one can attempt to proceed similar to the examples in the previous sections: For instance for simplicial (Regge) gravity one can construct, similar to the massless field, a refining based on piecewise flat geometry, at least on the classical level. Another possibility is to use some quasi-local notion of energy and to minimize this energy for the region that is being refined, in analogy to the procedure described in section 7.2.1.

215

7 Time evolution as refining, coarse graining and entangling pseudo gauge if diffeomorphism symmetry is broken [F25, F26, F66–F69]). But in general refining time evolution also adds physical (non–gauge) degrees of freedom. Thus refining time evolution leads to states that are supposed to be gauge equivalent, but seem to be based on different number of degrees of freedom. To resolve this puzzle, we need to understand states resulting from a refining time evolution as equivalent – they represent the same state on different discretizations. We will discuss quantum theory in section 7.3.1 in which this notion can be indeed made precise. For this interpretation it is important that the refined degrees of freedom are indeed in a state, that can be interpreted as vacuum. For instance in loop quantum gravity one uses the so–called Ashtekar–Lewandowski vacuum [F35, F36], in which spatial geometry is sharply peaked to be totally degenerate. An alternative vacuum state has been recently introduced [F38], in which the vacuum is rather peaked on flat connections. In both cases these vacua are used to define a notion of refining, in the second case this refining origins indeed from a time evolution of BF theory, a topological theory that describes flat connections. However both these choices are rather kinematical vacua, at least in 4D gravity ( [F38] gives actually the physical vacuum for 3D gravity). The interpretation as vacua is not tied to an energy functional, but rather to the fact that these states are the simplest possible ones from different viewpoints. The Ashtekar–Lewandowski vacuum can be understood as the state giving a constant value to all connection fields, whereas the BF vacuum [F38] gives a constant value to the conjugated variables, the flux fields describing spatial geometry. Thus, although these vacua are not physical, they show an important property of vacua (in homogeneous systems), namely to be homogeneous. We expect a physical vacuum to be given by a state satisfying the constraints and carrying a notion of homogeneity. Again the peculiarities of general relativity make this description of homogeneity non–trivial. Physical observables have to be invariant under space–time diffeomorphisms. To nevertheless allow for local observables one can use relation observables [F31–F34] , that often use a reference system built from matter (fields). Such a reference system requires, however, an inhomogeneity in these matter fields, which are used as rods and clocks. An alternative are reference systems built out of gravitational degrees of freedom, such as in [F100, F101]. There physical observables are constructed that describe perturbations away from homogeneity, thus a vacuum state can be described via a prescription for the expectation values and fluctuations of these observables. A different characterization of vacuum uses the notion of path integral as a projector on physical states. Assuming one can construct such a (consistent) projector we can define the image of any of the kinematical vacua under this projector as a physical vacuum. The kinematical vacua are homogeneous states – therefore one would expect the physical vacuum obtained by projection (assuming it can be constructed) also to be homogeneous. A priori it is not clear whether e.g. the Ashtekar–Lewandowski vacuum [F35, F36] and the vacuum, based on BF theory [F38] would lead to the same physical vacuum. As we will describe later, if we construct the projector on physical states via a refining time evolution operator (i.e. a path integral) such vacua can be also understood to realize the Hartle–Hawking no–boundary proposal for a vacuum [F86].

7.3 Refining in quantum theory Here we will discuss some aspects of refining time evolution in quantum theory. See also [F102, F103], which introduces a framework for time evolving Hilbert spaces, in which the number of degrees of freedom can increase and decrease. In this work we will base our discussion more on inductive limit Hilbert spaces and rather see refining time evolution as a means to define embedding maps needed for the construction of these inductive limit Hilbert spaces. We will explain this construction shortly in section 7.3.1. So far, this framework has been used on the kinematical level in loop quantum gravity. The main

216

7.3 Refining in quantum theory proposal of this work is that one should actually use the dynamics, that is time evolution, to define the embedding maps needed for this framework. With this proposal one adopts a no–boundary Hartle Hawking state as vacuum state, thus the vacuum (and a notion of equivalence between states of different refinement degree) is determined by the dynamics of the theory. This will be lined out in section 7.3.2.

7.3.1 Inductive limit construction of a continuum Hilbert space The inductive limit construction allows to define a continuum Hilbert space from a family of Hilbert spaces associated to discretizations (for instance graphs as in the Ashtekar Lewandowski representation [F35, F36] or triangulations as in the BF vacuum introduced in [F38]). The discretizations need to be organized into a directed partially ordered set, denoted by ({b}, ≺). The ordering provides a notion of coarser and finer discretizations, that is b ≺ b0 denotes that b0 is a refinement of b. In a directed partially ordered set one can always find a common refinement b00 for two discretizations b and b0 . We associate to each such discretization b a Hilbert space of states Hb . For any two Hilbert spaces Hb and Hb0 with b ≺ b0 , we need to define an embedding map ιbb0 : Hb → Hb0

.

(7.3.1)

These embedding maps have to satisfy consistency conditions: For any b ≺ b0 ≺ b00 we demand ιb0 b00 ◦ ιbb0

= ιbb00

.

(7.3.2)

As we will see, these conditions encode, under the identification of the embedding maps with time evolution maps, a path independence requirement of the time evolution maps. Given such a system, we can define the continuum limit of the theory, as an inductive limit. This limit is defined as the space of equivalence classes H := ∪b Hb / ∼. The equivalence relation is defined as follows: two states ψb and ψb0 0 are equivalent, if there exist a b00 with b ≺ b00 and b0 ≺ b00 , i.e. a discretization b00 refining both b and b0 , such that ιbb00 (ψb ) = ιb0 b00 (ψb0 ). In words, two states on different discretizations b, b0 are equivalent, if they can be refined to the same state. This notion of inductive limit allows to embed any ‘discrete state’ ψb into the continuum Hilbert space H via an embedding ιb . As mentioned this construction is used in loop quantum gravity on the kinematical level, that is the choice of embedding maps is not tied to a dynamics. Indeed, in a theory with a proper time evolution one would need to separate the refining time evolution steps into a ‘purely refining’ part and a ‘proper evolution’ part, as in the example in section 7.2.3. Otherwise one would identify time evolved states as equivalent. However, in gravitational theories, time evolution is a gauge transformation. In the quantum theory, the time evolution operator (7.1.1) is supposed to act as a projector onto physical states and thus as an identity on physical states. Hence one can indeed attempt to use refining time evolution, to define the embedding maps ιbb0 between different discretizations. As we will discuss, the difficulty is that refining time evolution maps based on ‘naive’ discretizations, will not satisfy the consistency conditions (7.3.2). Here coarse graining provides a means to reach theories in which the consistency conditions are actually satisfied.

7.3.2 Refining time evolution and no–boundary vacuum Let us return to the time evolution operator (kernel) defined from the path integral (7.1.1)   Z i K(Xini , Xf in ) = DX exp S(X) . (7.3.3) ~ Xini ,Xf in fixed

217

7 Time evolution as refining, coarse graining and entangling Here we denote by Xini and Xf in initial and final configuration data. In, for instance simplicial, discretizations of the path integral (7.3.3) the wave functions ψi (Xi ) and ψf (Xf in ) might be from two Hilbert spaces Hbi and Hbf associated to two different discretizations bi and bf . Even if we consider a system with proper time evolution the path integral (7.3.3) will project onto states satisfying the pre– and post constraints discussed in section 7.2. The reason is similar to the mechanism turning path integrals for gauge theories into projectors [F28, F29], we will sketch an argument here, valid for linearized theories [F104]: Consider for instance the case of post–constraints Ci (X, P ), where P are the momentum variables conjugated to X. We discussed in section 7.2 that these post–constraints are first class and lead to post–gauge degrees of freedom, that is part of the configuration data X at final time remain undetermined. On the other hand there will be also post–Dirac observables, i.e. functions on phase space that Poisson commute with the post–constraints C(X, P ). The structure of the constraints allows to make a canonical variable transformation such that the configuration variables separate into post gauge X G and post–Dirac X D degrees of freedom. The constraints then involve only variables conjugated to the post–gauge variables X G and the variables X G themselves. By integrating over all bulk variables in (7.3.3) we can define an effective action that only depends on initial and final configuration variables:   i K(Xini , Xf in ) = exp Sef f (Xini , Xf in ) (7.3.4) ~ We can use the canonical variable transformation for the final configuration data Xf in . The fact that the classical action leads to post–constraints means that the effective action decouples gauge and Dirac degrees of freedom Sef f = S D (Xini , XfDin ) + S G (XfGin ) .

(7.3.5)

This makes the appearance of constraints C(XfGin , PfGin ) obvious PfGin = −

∂S G (XfGin ) ∂Sef f = − ∂XfGin ∂XfGin

.

(7.3.6)

The time evolution kernel (7.3.4) is therefore of the form     i G G i D D K(Xini , Xf in ) = exp S (Xf in ) × exp S (Xini , Xf in ) ~ ~

.

(7.3.7)

All states resulting from a time evolution     Z i G G i D D ψf (Xf in ) = dXini exp S (Xf in ) × exp S (Xini , Xf in ) ψi (Xini ) (7.3.8) ~ ~   have a prescribed factor exp ~i S G (XfGin ) determining the dependence of the wave function in the gauge variables. Adopting a Schroedinger quantization scheme, with the momenta quantized as derivative operators Pˆ = ∂/∂X and configurations as multiplication operators, the states (7.3.8) satisfy the quantized constraints Cˆ = −i~

∂S G (XfGin ) ∂ + ∂XfGin ∂XfGin

.

(7.3.9)

In summary the choice of discrete action for a refining time evolution leads to constraints that determine the behaviour of the resulting wave functions in the ‘finer’ degrees of freedom, which here are characterized as post-gauge degrees of freedom.

218

7.3 Refining in quantum theory As mentioned this mechanism holds also for theories which a priori do not show any gauge symmetries and thus we deal with a proper time evolution operator. General relativity is a totally constraint system. Formal arguments show that the path integral (7.3.3) is equivalent to a projector onto the Hamiltonian and diffeomorphism constraints CI of the theory [F28, F29]   Z i Iˆ I . (7.3.10) DN exp N CI ~ Here N I denote Lagrange multiplier, known as lapse and shift. The integration over these multipliers induces a averaging over the action of the Hamiltonian and diffeomorphism constraints. For a discussion of the many subtleties involving this proposal see for instance [F30, F37, F83–F85].) The averaging would therefore project onto states that satisfy the constraints. In our discrete context, allowing for the possibility of discretizations changing in time, one expects that the Hamiltonian and diffeomorphism constraints will be part of the post– or pre– constraints. As mentioned this issue is however involved, as discretizations typically break diffeomorphism symmetry, which leads to the constraints. For the moment we will ignore this issue and comment later how to deal with it. Thus we can hope10 that a simplicial discretization of a path integral describing refining time evolution will lead to states which (a) satisfy the Hamiltonian and diffeomorphism constraints and (b) in which the finer (Dirac) degrees of freedom are also put into a specific state, characterized by the remaining post– constraints. With a simplicial path integral, we can in particular consider the extreme case of a refining time evolution; that is, we can start with zero-dimensional configuration space and evolve to a large triangulated spherical hypersurface [F10]. That is the first evolution step evolves from a vertex to the boundary of a d–dimensional simplex, where d denotes the space time dimension. The wave function will be just given as the (path integral) amplitude associated to this simplex. The following evolution steps can be understood as gluing further simplices to the one we started with, by multiplying the wave function with the corresponding simplex amplitudes and integrating over all variables that become bulk. In this case we will have at every step as many post constraints as (configuration) variables, i.e. the reduced phase space is zero–dimensional. Indeed all momenta Pb are generated by Hamilton’s principal function SH , i.e. the action evaluated on a solution prescribed by the boundary configurations X: P

=

∂SH (X) . ∂X

(7.3.11)

H We thus have constraints C = P − ∂S ∂X . These are Abelian, as the momenta are coming from a generating function. The phase space is foliated by gauge orbits, generated by the constraints, i.e. all configurations X are post–gauge degrees of freedom. In the quantum theory this corresponds to a unique physical wave function11 , given by a (Hartle Hawking) no–boundary vacuum [F86]. In the semi–classical approximation we have   i ψHH (X) ∼ exp SH . (7.3.12) ~

Here one would indeed expect the appearance of the standard vacuum, at least in the limit of infinitely large regions [F105]. Gravitational theories play a special role here, as the size of the 10

In fact, a naive discretization will break diffeomorphism symmetry and thus the statement regarding (a) can hold only in some approximate sense. 11 In fact, this wave function will in general depend on the underlying discretization, which can be interpreted as a choice of order for refining time evolution maps. Thus proper ‘uniqueness’ requires a notion of path independence, as will be explained in section 7.3.3

219

7 Time evolution as refining, coarse graining and entangling region is encoded in the state itself, thus the wave function gives rather a probability distribution for the geometrical volume of the hypersurface. ‘Radial’ evolution, as described here should not change physical states as it is just another form of time evolution. Thus we can hope that a vacuum is reached for degrees of freedom describing scales (much) larger than the discretization scale of the boundary. We note that a framework, which permits time evolution with phase spaces or Hilbert spaces that change in time, allows to define a notion of vacuum. For instance starting with a very coarse state and refining this state in an homogeneous manner should result into a state describing homogeneous geometries. This allows for applications for cosmology based on lattice treatments, for instance [F106–F108]. An interesting question for future research will be to investigate which simplicial quantum gravity models will lead to an acceptable (Hartle Hawking) vacuum and to investigate the properties of this vacuum. Apart from defining a no–boundary wave function, the refining time evolution can of course also be used to refine states – and thus to provide the embedding maps needed for the construction of inductive limit Hilbert spaces, as discussed in section 7.3.1. Such (dynamical) embedding maps are therefore selected by taking the dynamics of the theory into account, which is particularly advised for coarse graining [F9]. Here one has however to address the issue that discretized path integrals will in general break diffeomorphism symmetry and, related to this fact, be triangulation dependent. This will be subject of the next sections.

7.3.3 Path independence of evolution and consistent embedding maps We argued that a discrete evolution starting from a zero–dimensional phase space or a one dimensional Hilbert space produces a vacuum state. However this vacuum state will in general depend on the order of the time evolution steps, which for a simplicial discretization determines the triangulation of the bulk that is bounded by the triangulated hypersurface on which the vacuum is defined. Similarly, if we aim to use the refining time evolution defined by the path integral as embedding maps, the consistency conditions (7.3.2) will in general not be satisfied. These consistency conditions can now be interpreted as demanding independence of the evolved state from the chosen evolution path. It can be understood as a discrete version of implementing the Dirac algebra of (Hamiltonian and spatial diffeomorphism) constraints. As pointed out in [F109, F110] the Dirac algebra implies path independence, with respect to evolving through arbitrary choices of spatial hypersurfaces. This constitutes a further12 relation between diffeomorphism symmetry, that yields the constraints, and triangulation independence [F27]. So far we discussed only consistency for the embedding maps, which is needed to make the projective limit Hilbert space well defined. Observables on this Hilbert space need also to satisfy conditions known as cylindrically consistency: Observables Ob defined on the family of Hilbert spaces Hb need to commute with the embedding maps ιbb0 : ιbb0 (Ob ψb ) = Ob0 ιbb0 (ψb )

(7.3.13)

for all states ψb ∈ Hb and all pairs b ≺ b0 . This ensures that the observables are well defined on the continuum Hilbert space, i.e. do not depend on the representative ψb chosen. In the case that ιbb0 is given by a refining time evolution consistent observables have therefore to be ‘refining 12

In cases where diffeomorphism symmetry is realized, for instance in 3D discrete gravity, 4D gravity restricted to the ‘flat’ sector [F53], or 4D linearized gravity [F64], one can also introduce a continuum time evolution generated by Hamiltonian constraints [F24–F26, F111, F112] and define a (first class) Dirac algebra of these constraints [F113]. This continuum time evolution reproduces the discrete time evolution [F10,F24], if one integrates the infinitesimal evolution to one with a finite time.

220

7.3 Refining in quantum theory Dirac observables’. The algebra of refining Dirac observables characterizes the resulting continuum Hilbert space, as it provides a representation of this algebra. Topological theories can often be discretized such that partition functions and physical observables are triangulation independent. This also includes 3D gravity, which is topological. Due to the triangulation invariance the refining time evolution maps defined via the discretized path integral do satisfy the consistency conditions (7.3.2). We will illustrate this situation in section 7.6. As we will comment there, the set of ‘refining Dirac observables’ is much bigger than the set of (standard) Dirac observables given by the topological theory. This allows to use refining time evolution maps stemming from topological theories to define (kinematical) Hilbert spaces for other theories. They can also be used to construct a new Hilbert space for loop quantum gravity, based on the time evolution map of BF –theory [F38]. We believe that the application of refining time evolution maps is however not restricted to topological theories, despite the challenges posed by the triangulation dependence of the path integral. The strategy to attack this issue is to improve a given discretization by coarse graining. The fixed point of the coarse graining flow is hoped to show enhanced symmetry properties, in particular diffeomorphism symmetry which is tied to triangulation dependence [F27, F114]. Such a coarse graining flow leads however to non–local couplings13 , which are difficult to control. One would then also expect the embedding maps, if defined via refining time evolution, to be highly non–local. In section 7.4 we will discuss a coarse graining framework which avoids this issue, and moreover is based on the concepts introduced so far. Let us comment on the appearance of discretization changing time evolution in loop quantum gravity. There graph changing (actually graph refining) Hamiltonian constraints have been defined by Thiemann [F45–F47]. These constraints are anomaly free, in the sense that the commutator of two Hamiltonians vanishes if evaluated on the Hilbert space of diffeomorphism invariant states, see [F45–F47, F88, F89, F116, F117] for discussions. What is missing is a concrete geometric interpretation of the action of these constraints and a concrete connection to the path integral. (The notion of graph changing Hamiltonians inspired the development of spin foams, as time evolved spin networks [F118].) This discussion here suggest a possible interpretation for the graph changing Hamiltonians, the exponentiation of which should lead to a (refining) time evolution. Thus one could attempt to extract a notion of vacuum from the Hamiltonian constraints.

7.3.4 Pre–constraints and coarse graining We suggested to use the refining time evolution to define embedding maps for the inductive Hilbert space construction. Refining time evolution leads to post–constraints, which we argued characterize the (vacuum) state, into which the finer degrees of freedom are put. Here we want to comment shortly on the role of pre–constraints. These appear for coarse graining time evolution steps, that is the number of variables decreases. Classically these constraints demand that a state needs to satisfy certain conditions, so that the time evolution move can be applied. By time inversion symmetry we can understand this condition in the following way: the state has to be equivalent to a refining of a coarser state. Although the state is represented on a fine triangulation it does only include degrees of freedom in non–vacuum states on a coarser scale. Although for classical evolution one has to satisfy these constraints, quantum mechanical evolution is always possible. This also holds for standard gauge systems: a priori a quantum state does not need to satisfy any constraints to serve as a boundary condition in a path integral. Rather the path integral itself will project out non–physical degrees of freedom [F28, F29]. Thus, one can indeed expect that in a quantum evolution, the degrees of freedom which are too fine to be evolved 13

Triangulation invariant theories with local couplings are always topological theories, see for instance [F11, F115].

221

7 Time evolution as refining, coarse graining and entangling classically (as identified by the constraints) will be projected to the vacuum. In this sense the quantum mechanical evolution is automatically providing a coarse graining. Note that this will be a non–unitary evolution as it includes a projective part. A unitary description can only be obtained if one restricts to the subspace of the Hilbert space describing only sufficiently coarse degrees of freedom, i.e. the physical Hilbert space with respect to the pre–constraints, see also [F102, F103]. Thus time evolution cannot be inverted: Concatenating coarse graining and refining we isolate the projective part, that can be understood as projecting fine degrees of freedom to the vacuum state, that is the state obtained will automatically satisfy the initial pre–constraints. This provides an interesting asymmetry in time evolution that might serve as an arrow of time. See [F119] for another proposal on the origin of the arrow of time, in which also the notion of complexity of the state is crucial.

7.4 How to define a continuum theory of quantum gravity The investigation of time evolution with changing phase space or Hilbert space dimension is motivated by the simplicial discretization of gravity [F10,F11,F64]. However, such discretizations break diffeomorphism symmetry for the 4D theory [F24]. This appears both at the classical level [F25,F26] , and even in more severe form on the quantum level. For instance, the classical 4D Regge action, is invariant under 5-1 moves, but not under 3-3 moves. The latter fact can be related to a breaking of diffeomorphism symmetry on the classical level. Moreover on the quantum level one can show that no local path integral measure factor exists that makes the theory invariant under 5?1 moves [F120, F121], implying even a breaking of the residual classical symmetry. This implies in particular that the consistency conditions formulated in (7.3.2) are violated. A way out is to improve the discretization by coarse graining, see [F27, F49] for examples. At fixed points of the coarse graining flow one might arrive at perfect discretizations [F78, F79], for which consistency conditions of the form (7.3.2) are satisfied. These fixed points represent the continuum limit of the theory one started with, however expressed on a discretization. There are different ways to proceed with the coarse graining. One is to keep basic building blocks but to allow highly non–local couplings, which are naturally induced by the coarse graining [F80, F81]. As was pointed out in [F9], there is an alternative inspired by tensor network renormalization (which we will explain in section 7.5) and the generalized boundary proposal [F122]. This alternative construction of a consistent theory would not put basic building blocks (with simplest possible boundary discretizations) with their amplitudes in the centre but instead amplitude maps for space time regions, with arbitrarily complicated discretization of the boundary. These amplitude maps are built from the basic amplitudes, and agree basically with the (dual of the) Hartle Hawking no boundary wave function. The amplitude maps are defined on Hilbert spaces Hb associated to the discretized boundaries b of a space time region: Ab : Hb → C as   Z i Ab (ψb ) := DXDXb exp S(X, Xb ) ψb (Xb ) ~ =

hψ∅ |(K∅b )† |ψb i =: hψ∅ |ψb iphys

(7.4.1)

where we denote the bulk configuration variables with X and the boundary variables with Xb . Thus the amplitude map applied to the wave function ψb is given by the inner product between this wave function ψb and the no–boundary wave function. This no–boundary wave function is here expressed as time evolution operator K∅b applied to the one–dimensional wave function ψ∅ associated to the empty discretization. The second line in (7.4.1) defines the physical inner product, between the projections onto physical states of the two (kinematical) states ψ∅ and ψb . As usual the path integral in (7.4.1) is a discretized one. Thus the first task is to arrive at amplitude functionals Ab for fixed boundaries b that are independent of the bulk triangulation.

222

7.4 How to define a continuum theory of quantum gravity One way to reach such amplitudes is by coarse graining, as will be explained in the next section 7.5. For a very coarse boundary b we can triangulate the bulk with very few simplices. For instance the boundary of a simplex can be triangulated with just this simplex and thus the amplitude functional Ab , discretized in this way, would be just given by the pairing of the simplex amplitude with the boundary wave function. However there are infinitely many ways to subdivide this simplex (keeping the boundary), and thus one would have to find a method to determine the actual Ab . Indeed we will specify a further criterion for these amplitudes, which will actually help to construct the coarse graining flow of these amplitudes. It is important to note that bulk triangulation independence of the amplitude maps Ab is not sufficient for the construction of the continuum limit. (Indeed one could just declare some rule for selecting a particular bulk triangulation for each boundary.) We rather need to demand a condition that connects the amplitude maps Ab for different boundaries b. Thus we need first to choose embedding maps ιbb0 that connect the different boundary Hilbert spaces, as explained in section 7.3.1. As we explained, there might be different sets of embedding maps, leading to different continuum Hilbert spaces. We will see that some choices are preferred over others. With a given choice of embedding map we require that the amplitude maps are cylindrically consistent functionals, that is Ab0 (ιbb0 (ψb )) = Ab (ψb ) .

(7.4.2)

In words, if we take a coarse state and evaluate the corresponding amplitude map Ab on it, we should get the same result as first embedding the state into the ‘finer’ Hilbert space Hb and then evaluating with the ‘finer’ amplitude map Ab0 . Thus, the result should not depend on which boundary we choose to represent the equivalence class of states [ψb ] under the equivalence relations of the inductive limit. This allows to actually define the amplitude map as a functional on the inductive (i.e. continuum) limit Hilbert space H defined in section 7.3.1. Such a requirement was proposed in [F123] with regard to the (kinematical) embedding maps of the Ashtekar Lewandowski Hilbert space. We will argue here that the construction of cylindrically consistent amplitudes is facilitated by the adoption of dynamical embedding maps, as provided by refining time evolution. The amplitude map A[b] is technically not any more labelled by a discretization as such, but by equivalence classes of discretizations. Here two discretizations are equivalent if they can be refined to the same discretization. Thus, the information that is left over could just carry topological information (for gravitational theories where metric variables are dynamical) of the boundary. In our case we assumed spherical topology, thus a cylindrical consistent family of amplitude maps defines a continuum amplitude A. This amplitude A replaces the basic amplitude for, say the boundary of a simplex, one starts with in the regularization of the path integral. We can recover a ‘perfect’ amplitude, by evaluating A on states that are equivalent to states defined on a simplex boundary under the chosen embedding map. The cylindrically consistency requirement for the amplitude maps is a very strong requirement – it basically encodes the solution of the theory. We can hope to build such amplitude maps iteratively, for more and more refined boundaries, as will be the subject of the next section. To this end it is important to choose embedding maps that are adapted to the dynamics of the system [F9]. In particular we suggested that refining time evolution should give good embedding maps. A priori these will typically fail to satisfy the consistency requirement (7.3.2). However the improved amplitude maps Ab also allow to define an improved discretization of the path integral and thus to define a (refining) time evolution, that will satisfy the consistency requirement to a better approximations14 . In an iterative process one therefore improves both the amplitude maps and the 14

!

The consistency equations can be tested if one considers the equations on matrix elements hψb00 |ιbb00 (ψb )i = hψb00 |ιb0 b00 ◦ ιbb0 (ψb )i. Under an iterative improvement the equations will be satisfied for a larger and larger class of states involving finer and finer boundaries.

223

7 Time evolution as refining, coarse graining and entangling embeddings, if these are defined by refining time evolution. The reason why such embeddings are particularly apt to define cylindrically consistent amplitudes is in the definition of the amplitudes in (7.4.1). If ιbb0 = Kbb0 we will have Ab0 (ιbb0 ψb ) = hψ∅ |(K∅b0 )† |Kbb0 ψb i ∼ hψ∅ |(K∅b )† |ψb i = Ab (ψb )

.

(7.4.3)

Here we wrote ∼ in the second equation as (K∅b0 )† ◦ Kbb0 ∼ (K∅b )† holds only approximately in the discretization. However we see that embedding maps defined via refining time evolution simplify the task of constructing cylindrically consistent amplitudes. Indeed the consistency condition for these embedding maps are tied to the cylindrical consistency of the amplitudes. We want to remark that we do not require a consistent gluing between the cylindrically consistent amplitude maps as long as this gluing is performed on a discrete boundary b. (That is the gluing involves integration only over the variables Xb .) One could for instance require that the amplitude for a region with a more complicated boundary Ab3 arises as the gluing between two amplitudes with less refined boundaries Ab1 and Ab2 , similar to the way one would glue simplices together. We expect that such a relation might indeed hold, however only if one preforms a continuum limit for the piece of boundary that is glued over. Let us emphasize that the amplitude maps A[b] are the end point of a construction to reach the continuum limit of the theory. Of course one hopes that the ‘initial’ theory defined via basic building blocks and local couplings, provide the basis for the construction of such a theory. This implies that this ‘initial’ theory can nevertheless be used to extract sufficiently coarse grained observables from sufficiently refined discretizations. As mentioned we aim at constructing both, cylindrically consistent amplitudes and consistent embedding maps given by refining time evolution. In the next section 7.5 we will explain that tensor network coarse graining tools provide methods to construct these.

7.5 Tensor network coarse graining: time evolution in radial direction Tensor network renormalization group methods [F7,F8,F124–F127] can be understood to implement an iterative method to construct cylindrically consistent amplitudes. Coming back to equation (7.4.3) Ab0 (ιbb0 ψb ) = hψ∅ |(K∅b0 )† |Kbb0 ψb i ∼ hψ∅ |(K∅b )† |ψb i = Ab (ψb )

.

(7.5.1)

we can understand the second term to consist of two parts: the first is the computation of hψ∅ |(K∅b0 )† , that is the basically the amplitude functional Ab0 for a more refined boundary. One would build such an amplitude functional from gluing amplitudes Ab for less refined boundaries b. However we want to define an iterative process that improves the amplitude maps Ab , which are functionals on Hb . We thus have to find a way to pull back the amplitudes Ab0 to Hb , which is done by using the embedding map ιbb0 = Kbb0 . Thus one defines the improved amplitudes Aimp as b † Aimp b (ψb ) = hψ∅ |(K∅b0 ) |Kbb0 ψb i

.

(7.5.2)

Here both (K∅b0 )† and Kbb0 are built from using the initial Ab as basic amplitudes. The process is repeated for the improved amplitudes Aimp until the procedure converges to a b f ix fixed point Ab . This fixed point amplitude can be used to proceed to a more refined pair of boundaries (b0 , b00 ) with b ≺ b00 to find the next fixed point amplitude Afb0ix and so on. There are many tensor network renormalization algorithms [F7, F8, F124–F127], which differ in their geometric setup and the details of how to define (K∅b0 )† and the embedding ιbb0 . We will shortly explain a method that can be interpreted as radial evolution, as this also matches nicely with amplitudes being defined via the no–boundary wave function, as in (7.4.1).

224

7.5 Tensor network coarse graining: time evolution in radial direction

•

•

•

•

•

•

•

•

•

•

Figure 7.6: Illustration of radial time evolution in tensor networks: By adding eight additional tensors (in gray) we perform one time evolution step. The boundary data grows exponentially from χ4 to χ48 .

The name tensor networks refers to the fact that the amplitudes of a space time region are encoded in tensor of a given rank n, associated to an n–valent vertex, which we can imagine to sit inside this space time region. The indices of this tensor encode the boundary data of the space time region, hence contracting tensors of two neighbouring regions corresponds to gluing the associated amplitudes. The rank n and the bond dimension χ (equal to the number of values the index can take) determines the amount of boundary data and hence the fineness of the boundary in question. Note that one can redefine higher rank tensors to tensors of lower rank by summarizing for instance two indices (i, j) with χi , χj into effective indices I = (i, j) with bond dimension χI = χi · χj . This interpretation matches nicely with spin nets [F128] and spin foams describing gravitational dynamics. The former can be naturally understood as tensor networks [F2, F3, F129, F130]. A tensor network description of spin foams can be found in [F2, F3].

7.5.1 Radial evolution As we will see tensor network algorithms are related to transfer matrix methods in which the (Wick rotated) time evolution operator is diagonalized. For the latter, Wick rotation is essential, as the the eigenvalues of the transfer operator need to be ordered in size; in this way we can distinguish relevant from irrelevant degrees of freedom. However one can understand tensor networks to replace the time evolution operator with a radial evolution operator. Even if the (standard) time evolution operator might be unitary, and hence all eigenvalues with absolute value equal to one, the radial evolution operator will include a projective part, that – as we have argued will project out finer degrees of freedom. This can then be used for the construction of an embedding map. An evolution in radial direction is also expected to project onto the vacuum state [F7, F105], see also the discussion in section 7.3.2 which involves the non–Wick rotated amplitudes.15 Consider a radial evolution as in figure 7.6. Here the amplitude / tensor for a larger region is built from the amplitude/ tensor of a basic building block, represented by a (dual) vertex. One would now like to treat the amplitude for the new region as an effective tensor and repeat the procedure. However one has to face the problem, that the number of boundary data, grows exponentially during this procedure, making it impossible to implement in practise. We thus have to find a method to project back the amplitudes to a boundary with less data, i.e. to coarser boundaries. The radial time evolution can be split into steps with time evolution operators Z R2 T (R1 , R2 ) = exp(− Hr dr) . (7.5.3) R1

This is a refining time evolution in the sense that the Hilbert space H2 at R2 will have more 15

RR For a (Wick rotated) time evolution operator exp(− 0 Hr dr) acts as a projector on the ground states of the Hamiltonian H for R going to infinity. Here Hr denotes the Hamiltonian for radial evolution at the radius r. For large radius, we will have small dr/r, and hence Hr approaches the Hamiltonian H for the time evolution of constant volume hypersurfaces.

225

7 Time evolution as refining, coarse graining and entangling

A

.. .

M

α β

M

.. .

B

A

.. .

M

α β

•

V

i

Figure 7.7: Left: Two regions in a tensor network, encoded in the matrices M , are sharing two edges with labels {α, β}, which have a total range of χ2 . Right: From the singular value decomposition we can define the map V depicted as a three–valent vertex, where we restrict the label i of the singular values to be ≤ χ.

kinematical degrees of freedom than the Hilbert space H1 at R1 . Thus T (R1 , R2 ) will have a projective part (even if we would not have Wick rotated), which can be identified by a singular value decomposition. This would give a maximal number of dim(H1 ) singular values. Hence T (R1 , R2 ) will have a non–trivial co–image in H2 , which can be projected out. A new amplitude can therefore be defined on a dim(H1 ) subspace, which can be identified as the subspace carrying coarse boundary data. Such a scheme might be indeed worthwhile to investigate further (for statistical systems), in order to obtain an intuition about the truncations. One would however expect that the reorganization of the degrees of freedom via the singular value decomposition will be highly non–local, as we have seen for the example in section 7.2.1. The time evolution considered there exactly corresponds to T (R1 , R2 ). This non–locality makes it however difficult to turn this into an iterative procedure. The new amplitudes will be expressed with respect to data spread over the entire boundary, which makes a local gluing of these amplitudes difficult.

7.5.2 Truncations via singular value decomposition In practice one therefore employs schemes which involve more local truncations. The basic idea is as follows. Imagine two space time regions or effective vertices connected with each other by two edges, representing the summation over a certain set of variables, see figure 7.7. We would like to replace these edges carrying an index pair {α, β} of size χ2 with an effective edge carrying only a number χ of indices. We choose an optimal truncation for the summation over the index pair {α, β}, which is given by the singular value decomposition of MAαβ : 2

MAαβ =

χ X

UAi λi Vi αβ

(7.5.4)

i=1

where λ1 ≥ λ2 ≥ . . . ≥ λχ2 ≥ 0 are positive, and U, V are unitary matrices. The truncation consists then in dropping the smaller set of singular values λi with i > χ. Pictorially Viαβ restricted to i ≤ χ defines a three–valent vertex and we can use these three–valent vertices as in figure 7.7 to arrive at a coarse grained region with less boundary data.

7.5.3 Embedding maps and truncations We can understand the tensors V as coarse graining maps. Alternatively, if read in the other direction, these maps provide the embeddings ιbb0 from a coarser to a finer discretization. This interpretation comes from seeing the partition function (with boundary) as a functional (or amplitude map) Ab on a ‘boundary’ Hilbert space Hb . Gluing several space time regions together we obtain a partition functional A0b0 which a priori acts on a Hilbert space Hb0 with finer boundary, see figure 7.8. We can however pull back this functional with the embedding map defined via the

226

7.5 Tensor network coarse graining: time evolution in radial direction tensors V and in this way obtain an effective amplitude map A0b : A0 b (ψb ) := A0 b0 (ιbb0 (ψb )) .

(7.5.5)

This gives a renormalization flow for the amplitude maps, and a fixed point is reached if A0 b = Ab .

A0b0

ψb0

A0b

ψb

Figure 7.8: To obtain an effective amplitude map A0b on the coarser boundary b one can pull back the amplitude A0b0 from the finer boundary b0 via the previously defined embedding maps.

We see that it is essential to construct three–valent vertices (with its associated tensors), which we can see as special cases of coarse graining or refining maps. These three–valent vertices should be adapted to the four–valent ones giving the ‘regular’ dynamics. They can be understood to give a coarse graining or refining version of the regular evolution defined by the four–valent vertices. Note that this adaptation has to happen after each of the coarse graining steps, as the four–valent tensors flow under coarse graining. The singular value decomposition (or generalizations as in [F125–F127]) provides one method to construct such three–valent tensors. Interestingly, geometric theories such as, spin foams [F14] or spin nets [F2, F3, F128] provide already descriptions for vertices of arbitrary valence [F131–F133]. Thus one can imagine a lattice of say four–valent and three–valent vertices, which automatically implements the coarse graining procedure. However, if one believes that these vertices provide good truncations, in the sense of approximating the summation over an index pair well by a summation over just one index, one needs to adapt the three–valent vertices to the dynamics encoded in the four–valent vertices. That is the embedding maps have to flow together with the effective amplitude maps. Such a relation is provided by the singular value decomposition in (7.5.4). Alternatively, spin foam construction tools [F130–F133] provide methods of how to build vertices of arbitrary valency out of a given vertex. This construction has also to be performed at every step of the coarse graining. It will be interesting to see, whether such a method gives a good truncation. The advantage of such a method is that it might be far easier to implement for spin foams, than the singular value decomposition, and might lead to a closed flow equation.

7.5.4 Embedding maps for the fixed points Ultimately one would expect that the relation between vertices of different valencies is just given by gluing, i.e. a four–valent vertex is given as a gluing of two three–valent vertices. Indeed this relation can be obtained from the singular value decomposition (7.5.4) in case that (a) all non– vanishing singular values are equal to one and (b) if one is working in a truncation, the number of non–vanishing singular values needs to be smaller than χ. These conditions are satisfied at the stable fixed points of the renormalization flow (describing the phases of a given system), see for instance [F129]. Condition (a) is expected to arise in theories with diffeomorphism symmetry – where time evolution is a projector. (Again the problem is that diffeomorphism symmetry is broken under discretization, so the projector property does not hold exactly and might be rather expected to emerge after sufficient coarse graining. For a computation of the transfer matrix in spin foam theories and a discussion whether these are projectors, see [F128, F134, F135].)

227

7 Time evolution as refining, coarse graining and entangling Condition (b), in case that one is working with a cut–off, basically imposes a topological theory for fixed points that are triangulation invariant16 . For instance fixed points identified in [F2, F3, F129, F130] via a tensor network coarse graining describe triangulation invariant and therefore topological models and χ gives the maximal number of propagating degrees of freedom. Indeed we will see in section 7.6 that all the proposals outlined here are explicitly realized. For an interacting theory, such as 4D gravity, one would expect to need an infinite bond dimension χ, as indeed arises around phase transitions. With a fixed χ one can however approach the phase transition up to a certain precision, and, as the method is designed to keep the variables describing coarse excitations, one can hope to obtain reliable predictions for sufficiently coarse observables. In light of the previous discussion this means to obtain amplitude functionals, which satisfy the cylindrical consistency conditions sufficiently well for coarse boundaries.

7.6 Topological theories The previously introduced and discussed concepts of time evolution via coarse graining / refining and the concept of cylindrically consistent amplitudes are perfectly realized in topological field theories. In the following we would like to emphasize a few key points. In the first part of this section, we will mainly refer to topological lattice field theories in 2D, for instance [F136]. For theories with a geometric interpretation, see for instance [F115, F130]. Consider a 2D lattice topological field theory with partition functions defined on three–valent graphs. As for a three–valent tensor network we have weights or tensors associated to the vertices and variables and hence Hilbert spaces associated to the edges. The partition function is then defined by summing the variables associated to bulk edges. In case of a boundary, we keep the corresponding variables fixed, thus obtaining a partition function depending on these boundary values. Alternatively we can understand the partition function as an operator (between two boundary Hilbert spaces) or a function (on one boundary Hilbert space). In this vain a three–valent graph (7.6.1) represents the simplest graph and the associated partition function, interpolating between a two– site boundary Hilbert space on the lower boundary and a one–site boundary Hilbert space on the upper boundary. In this case we can understand this partition function as a (coarse graining) time evolution map, not involving any bulk summation. The same holds for the time inverted vertex. Topological lattice theories are triangulation invariant (referring to the triangulation dual to the graph). Thus the partition function does only depend on the topology of the manifold and not on the choice of the triangulation. (We will later show that this also holds in a certain sense for the boundary triangulation, due to the cylindrical consistency of the partition function.) Pictorially this corresponds to the following equalities = c

,

= c

,

=

.

(7.6.2)

Here we assume always a summation over the variables or indices associated to the bulk edges. The equations have to hold for all possible choices of the boundary variables. In this section we will assume that the constant c is actually finite and hence can be adjusted to c = 1 by a rescaling of the amplitudes. 16

Rank three tensors can be found from a (singular value) decomposition of the rank four ones. These tensors are associated to three–valent vertices dual to triangles, we can therefore consider models on irregular triangulations. The fixed point condition for coarse graining on a regular lattice are then weaker than requiring triangulation invariance.

228

7.6 Topological theories Given the 2 − 2 move invariance, we can replace the 3 − 1 move by the so–called bubble move j = c j

.

(7.6.3)

j The equivalence of bubble and 3 − 1 move (given the 2 − 2 move holds) follows from the following calculation =

= c

.

(7.6.4)

7.6.1 Time evolution as coarse graining and refining Using this pictorial representation we can symbolize a local time evolution operator or transfer matrix acting on a two site boundary Hilbert space as follows: T =

=

X

ED λi ι(i) ι(i)

.

(7.6.5)

i

Time is flowing upward. From the bubble move we see that the time evolution operator is actually a projector: T2 =

=

X i,j

λi λj

E ED D X E D λ2i ι(i) ι(i) = ι(i) ι(j) ι(i) ι(j) = | {z } i

= T

.

(7.6.6)

=δi,j

Hence the eigenvalues are λi = 1 ∨ λi = 0. The construction of dynamical embedding maps via a singular value decomposition is trivial and it is straightforward to split the projector into two maps, one that can be interpreted as a coarse graining, the other in terms of a refining: C :=

,

R :=

.

(7.6.7)

Each of these maps can be interpreted as maps between Hilbert spaces of different dimension. Concatenating the two gives either the time evolution operator back or is just the identity: This tells us both that the refined state does not carry additional information and that no (physical) information is lost under coarse graining. We will extend this to arbitrarily large triangulations below. The time evolution operator acts locally, such that it is possible to only locally evolve a state in time, e.g.: .

(7.6.8)

However in which order one time evolves pairs of lattice sites is an arbitrary choice, one which should not influence the results of the theory such as the partition function. Therefore we impose the following consistency condition, which is satisfied in topological field theories and allows us to define the transfer matrix for three discretization sites uniquely:

=

=

.

(7.6.9)

229

7 Time evolution as refining, coarse graining and entangling This construction can be generalised to arbitrarily many discretization sites. In all cases we can replace the time evolution map with a graph of the form on the right hand side of equation (7.6.9). Thus the maximal rank of this time evolution map (which is a projector) is given by the bond dimension of the kink in the middle, i.e. the bond dimension of one edge. This gives also the maximal number of physical degrees of freedom.

7.6.2 Consistent embedding maps and inductive limit Furthermore if one cuts this diagram into two pieces at the kink, one obtains both more general coarse graining and refining maps. Hence the consistency conditions naturally translate to both the coarse graining and refining maps, as we demonstrate for the refinement map R: =

=

.

(7.6.10)

Thus these refinement maps satisfy the path independence conditions as outlined in section 7.3.3, and therefore allow the construction of an inductive limit of Hilbert spaces as described in section 7.3.1. This gives the continuum limit of this theory. Furthermore, understanding the partition function Ab (with boundary b) as a functional on the boundary Hilbert space Ab : H b → C

,

(7.6.11)

we also obtain that the partition function is a cylindrically consistent operator [F115]: Ab0 (ιbb0 (ψb )) = Ab (ψb )

(7.6.12)

(which coincides with the fixed point condition (7.5.5)). Here ιbb0 is built from the refinement maps R, in the way described above. Given a boundary b we choose a triangulation (or dual three–valent graph) interpolating this boundary. As long as we choose a fixed topology for this interpolating triangulation, the partition function will not depend on this choice and hence is a well defined functional on the boundary Hilbert space. Additionally we can refine the boundary via a refinement move. An interpolating triangulation can be obtained by just including a coarse graining move at the appropriate dual edge. This will give a ‘bubble’ that can be removed due to the bubble move invariance, and we arrive at the previous partition function acting on the unrefined Hilbert space. Hence the partition function (actually a family of functionals labelled by the boundaries b) is cylindrically consistent with respect to the embeddings provided by the refining time evolution. That automatically allows to define from the (so far) discrete partition functions a continuum limit on the projective limit Hilbert space. This is to our knowledge a new insight, as topological theories are often only discussed with regard to the invariance of the bulk triangulation. The refinement maps can also be used to construct the Hartle Hawking vacuum states mentioned in section 7.2. To this end one has either to dualize one edge (graphically a bent or cup). Alternatively in examples where the edges are labelled by (SU (2)) spins, we start with a refining map for which we fix on the incoming edge j = 0, which gives the Hilbert space C associated to this edge. The (two–site) Hilbert space can then be refined further in an arbitrary way, giving the Hartle Hawking vacuum state on boundaries with different numbers of sites. Doing this in a linear way, i.e. as for the graph on the right in equation (7.6.10) gives matrix product states (MPS) [F137,F138]. MPS provide ans¨atze for ground state wave functions of Hamiltonians. The projectors T defined above correspond to exponentiated Hamiltonians and the type of MPS defined in (7.6.10) is the ground state to the following Hamiltonian: ! N −1 X , (7.6.13) H= I− I=1

230

I

7.6 Topological theories where I denotes the index pair the projector is acting on for a total of N outgoing legs. Thus Hartle Hawking vacuum states appear here as the ground states of the Hamiltonians (7.6.13), justifying again the notion of these states as vacuum states.

7.6.3 3D topological theories and entangling moves Similar statements hold for higher dimensional theories, for instance BF theories. There are however interesting differences pertaining to the role of Pachner moves in discrete topological theories based on triangulations, such as the Turaev–Viro models [F139]. The physical states of these models can be described as string net states [F140]. For a canonical time evolution in (2+1)D we will have 3 − 1, 1 − 3 and 2 − 2 moves as time evolution moves as described in section 7.2. These allow to build an arbitrary complex triangulated hypersurface from a simple one. In this way one can build up an analogous MPS representation of a string net state on arbitrary complex 2D triangulations or on the corresponding dual graphs. The 3 − 1 and 1 − 3 moves serve as (purely) coarse graining or refining moves, whereas the 2 − 2 moves (dis–)entangle the degrees of freedom. The latter play an important role in entanglement renormalization [F40, F63]. Interestingly MPS states in one (spatial) dimensions do not lead to long range entanglement [F141, F142], whereas the example just described gives a phase with long range entanglement in two (spatial) dimensions [F63]. This might be due to the necessity of the 2 − 2 move to obtain triangulations not equivalent to a stacked sphere (which would not support long range entanglement). In the case of the stacked sphere the consecutive 1 − 3 moves can be represented by a tree graph. In (1+1)D all triangulations of a circle can be obtained as ‘stacked spheres’, which are dual to trees. In (3 + 1)D we have similarly 1 − 4 and 4 − 1 moves as refining and coarse graining moves respectively. As mentioned before the 4 − 1 move does not add physical degrees of freedom, as all additional degrees of freedom are associated to Hamiltonian and diffeomorphism constraints [F10,F25,F26]. Additionally we have 2 − 3 and 3 − 2 moves, which can be interpreted as entangling moves, similarly to the 2 − 2 move in (2 + 1)D.

7.6.4 Constructing inductive limit Hilbert spaces for non–topological theories We discussed that the embedding maps provided by the refining time evolution of topological theories satisfy the consistency conditions (7.3.2). Thus one can use these embeddings to construct an inductive limit Hilbert space as outlined in section 7.3.1. Note that this Hilbert space will support a much bigger class of observables than just the Dirac, i.e. topological observables of the topological theory. The set of observables supported by this Hilbert space is determined by the cylindrical consistency conditions (7.3.13) for observables. The cylindrical consistent observables then describe excitations from the vacuum state, which is given by the no–boundary wave function of the topological theory. Thus the excitations in particular violate the constraints (implementing the equations of motion) of the topological theory. This is a general proposal for the construction of inductive Hilbert spaces. It will be interesting to explore more in detail the relation between the set of cylindrical observables which characterize the inductive Hilbert space and the topological theory which provides the embedding maps. This strategy to construct an inductive limit Hilbert space has been realized recently [F38] for the topological BF theory, which describes the moduli space of flat connections. In this case the excitations are parametrized by curvature observables. The set of cylindrically consistent observables is given by a holonomy–flux algebra underlying the formulation of loop quantum gravity and (lattice) gauge theories. This method therefore resulted in an alternative representation and a new vacuum for loop quantum gravity.

231

7 Time evolution as refining, coarse graining and entangling

7.7 Geometric interpretation of the refining maps Here we wish to point out that geometric theories are very special with regards to coarse graining and refining.17 This is due to the fact that the geometry itself is included into the set of dynamical variables. The (say semi–classical) state on the boundary of a region determines a geometry for the bulk, defined as solution of the Einstein equations for the given boundary data the state is peaked on. (This of course assumes that one has a sensible theory of quantum gravity, which would result in a semi–classical state for the Hartle Hawking state.) Thus setting (geometric) scale and number of coarse graining steps as equal is, at least a priori, senseless in such theories. Rather, a renormalization scale is given by the coarseness or fineness of the boundary data, that is, the scale on which geometric properties, such as curvature, vary. Even if one peaks the boundary state on a given geometry with a fixed (hypersurface) volume one cannot expect to find that the partition functions peaks on some regular bulk geometry such that the bulk volumes are bounded by the hypersurface volume. The reason is that one expects diffeomorphism symmetry to emerge in the form of vertex translation invariance. This symmetry even allows to move the vertices such that orientations of building blocks are inverted. This corresponds to ‘spikes’ in the geometry, see for instance [F51, F144]. These spikes give rise to divergences [F23, F50–F52], related to the non–compactness of the diffeomorphism gauge orbits. As we will argue below this mechanism allows the appearance of arbitrarily large spins even in a region bounded by a small boundary geometry. This might make even a theory describing flat geometry, such as 3D BF , appear as highly fluctuating. However (almost) all these fluctuations are gauge fluctuations [F50], due to the diffeomorphism gauge symmetry. We will illustrate this with a 2D example below. The relation between the sum of orientations and divergences has been pointed out in [F144] which also argues that allowing only one orientation could cure the problem of divergences. However, we will show here, that from the perspective of time evolution as a refining and coarse graining map, the appearance of two orientations is very natural. (It is also natural as the gravitational constraints are quadratic in the momenta describing time symmetric evolution. The two solutions of the quadratic equation correspond to the two orientations.)

7.7.1 2D example Let us illustrate this with the intertwiner models introduced in [F115]. These host families of topological theories, which have all the properties discussed in section 7.6. Moreover the theories allow for a natural geometric interpretation, as they are defined on three–valent graphs. The edges carry a spin j (SU (2) representation) and a magnetic quantum number. The spin can be interpreted as a length variable – indeed at the three–valent vertices triangle inequalities have to be satisfied arising from SU (2) recoupling theory. Consider one upward pointing line as in the examples before: This line can be interpreted as a line in a (2D) space with a length given by the spin j. Using our previously defined refining maps R, we can map it to a different state, which is now labelled by two spins j 0 and j 00 :

j

→ →

17

X j0 j

j 00 (7.7.1)

j ,

(7.7.2)

See also the discussion in [F143], which argues that in reparametrization invariant theories all couplings are dimensionless.

232

7.7 Geometric interpretation of the refining maps where the spins j 0 and j 00 have to satisfy triangle inequalities. However this picture is indistinguishable from adding a triangle with opposite orientation and hence ‘removing’ it: →

j

. j0

(7.7.3)

j 00

Thus for a quantum theory this means that both possible orientations have to be taken into account in a superposition: ∼

⊕

.

(7.7.4)

This picture conforms with refining the edge and adding a vacuum degree of freedom (this degree of freedom is not physical, as we are considering a topological theory here): This vacuum degree of freedom allows fluctuations of the edge geometry around a flat subdivision – in the sense that the refined edge can bend upwards or downwards. Any asymmetry would appear as proper time evolution, which we do not expect for a topological or gravitational theory. In this case time evolution is generated by constraints and hence gauge – and as explained before realized as a projector in the quantum theory. Note that the fluctuation can be arbitrarily large, as argued below. In this case this can be linked to diffeomorphism symmetry realized by a vertex translation symmetry. The middle vertex can be translated an arbitrary large distance forward or backward (or sideways) in ‘time’. Moreover, as this is a gauge symmetry, all such configurations have to be gauge equivalent, i.e. come with the same amplitude (and a diffeomorphism invariant measure18 ).

7.7.2 Spin foams A similar picture applies to spin foams, where gluing a simplex to a boundary can be done with two orientations. From the semi–classical expressions for the simplex amplitude we again obtain a geometric picture: i.e. with a 1 − 4 move (from gluing a 4–simplex to one boundary tetrahedron) we replace a boundary tetrahedron with a complex of four tetrahedra, that now allows the inner geometry of the original tetrahedra to fluctuate around a flat subdivision. Note that also in this case one did actually not add a physical degree of freedom, at least not if one deals with Regge geometries [F10,F53,F64]. The reason is that the new kinematical degrees of freedom (the four new edge lengths) are accompanied by four (Hamiltonian and diffeomorphism) constraints, associated to the new vertex. These allow to move the additional vertex arbitrarily forward or backward in time, explaining the appearance of the two orientations. As before the diffeomorphism or vertex translation symmetry also means that configurations with arbitrarily large length of the four inner edges have equal amplitude to those describing a ’flat’ subdivision, thus one would expect divergences to appear for every inner vertex, for a discussion in spin foams see [F23, F50–F52, F145]. Thus one should be very careful with treating the spin j variable, which encode the length or area variables in 3D or 4D respectively, as an order parameter. (Indeed one should consider diffeomorphism invariant observables as order parameters, which are however hard to come by [F31–F34].) We can provide here an interpretation of the divergences as coming from (extremely) squeezed states: As mentioned we add only degrees of freedom in the vacuum state (including gauge degrees of freedom), in the case of 4 − 1 moves these have to satisfy the Hamiltonian and diffeomorphism constraints. Thus, fluctuations in the ‘constraint’ directions are completely suppressed, whereas fluctuations in the conjugated (i.e. gauge) directions become infinitely large, represented as a non– compact gauge orbit of configurations with equal weight. 18

This actually allows to determine the path integral measure, see [F120].

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7 Time evolution as refining, coarse graining and entangling

7.8 Discussion We pointed out that gravitational theories in a simplicial description, provide with their time evolution maps automatically refining, coarse graining and entangling maps. More generally we interpret the degrees of freedom added during refining time evolution moves as degrees of freedom in the vacuum state (or gauge degrees of freedom). This suggests the construction of a global vacuum state as a state evolved from a one–dimensional Hilbert space C, see also [F10, F102, F103], which gives a simplicial realization of the no–boundary proposal [F86]. Indeed via the notion of dynamical cylindrical consistency [F9], we can identify the vacuum states as representing the equivalence class which includes the unique state in the ‘no–boundary’ Hilbert space C on different discretizations. We argued that the time evolution maps provide embedding maps for the construction of the continuum limit via projective techniques. In section 7.4 we outlined how to define and arrive at a consistent continuum dynamics for quantum gravity. This is based on the dynamical embedding maps and proposes to construct the amplitude maps as cylindrically consistent maps based on these embeddings. This allows to define the amplitude maps as objects of the continuum theory on the continuum Hilbert space. Such (dynamical) embedding maps have however to satisfy stringent path independence conditions, which we related to the path independence under different choices of interpolating hypersurfaces [F109, F110] and an anomaly free representation of the Dirac algebra of constraints [F45–F47, F113]. These conditions are indeed hard to satisfy exactly for interacting theories but should be valid in some approximate sense if considering sufficiently coarse grained observables. We explained that tensor network renormalization algorithms provide a method to construct dynamical embedding maps that do satisfy the consistency conditions to a better approximation and the related (approximately) cylindrically consistent amplitudes. An important ingredient in these algorithms are truncations. Good truncations are basically good reorganizations of the degrees of freedom into coarser ones and finer ones. We argued that such a splitting can be found by employing radial, that is refining, evolution. In topological theories the refining time evolution maps typically satisfy the path independence conditions. This allows the construction of projective limit Hilbert spaces using refining time evolution as embedding maps. This will realize the physical state of the topological theory (satisfying the constraints of the topological theory) as a vacuum in this projective limit Hilbert space. This vacuum coincides with the no–boundary wave functions. Excitations can be produced by cylindrically consistent observables. An example of this construction has been recently provided in [F38]. For non–topological theories, such as 4D gravity, we suggest that an exact satisfaction of the path independence conditions for the embedding maps would rather involve non–local dynamics, as is indicated by the discussion in section 7.2.3. The construction of the continuum limit in section 7.4 allows for such non–local embeddings. The necessity of a non–local dynamics has been recently argued for in [F121], which points out that linearized 4D quantum Regge calculus requires a non–local path integral measure in order to show invariance under 5 − 1 moves. From a statistical physics point of view one would expect that a second order phase transition is needed for the continuum limit, leading to long range (in terms of number of lattice cites) correlations and a conformal theory at the boundary. Indeed in the context of tensor network algorithms and radial evolution the appearance of a conformal theory at the fixed point leads to an interpretation in terms of AdS geometries and holographic renormalization, for instance [F41–F43]. For the case of non–perturbative gravity such an interpretation might not apply straightforwardly. Here one would expect that the boundary variables or the quantum state defined on the boundary encodes the geometry of the boundary and – via the equations of motion – of the bulk. There are still many puzzling features to explore in the context of discretization changing time evolution. This in particular applies to interacting theories, such as 4D gravity. As we outlined here such discretization changing evolution might however provide a definition of the physical vacuum and more generally allow the construction of the continuum limit of the theory. This makes the

234

7.8 Discussion explorations of these issues very worthwhile.

Acknowledgements B.D. thanks Philipp H¨ ohn for collaboration, intensive discussions and for providing a draft version of [F102, F103], as well as Kyle Tate for collaboration on [F77]. We thank Wojciech Kaminski and Mercedes Martin-Benito for collaborations and discussions and Guifre Vidal for pointing out reference [F63] and explaining the (corner transfer) tensor network algorithm to us. B.D. would like to thank Laurent Freidel, Marc Geiller, Ted Jacobson, Aldo Riello and Lee Smolin for discussions. We thank the referee for helpful comments, which led to an improvement of the manuscript. S.St. gratefully acknowledges support by the DAAD (German Academic Exchange Service) and would like to thank Perimeter Institute for an Isaac Newton Chair Graduate Research Scholarship. This research was supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

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8 Discussion The overarching question of this thesis concerns the intricate relation between discrete models of quantum gravity and the classical continuum theory of general relativity, which we explore at very different levels of the discretisations. Some presented results only deal with one building block, while in others we develop and apply tools to study the collective behaviour of literally infinitely many building blocks and discuss how to consistently construct a continuum theory from the discrete one. At first sight, these results appear to be partially disconnected or isolated. Therefore, the purpose of this discussion is not only to discuss their significance with respect to the overarching questions and how they fit into the already existing knowledge in quantum gravity, but to link them together to form a consistent line of argument. As a guiding principle we will order the discussion of the results by the number of involved fundamental building blocks, starting only from a few moving towards the limit of infinite refinement. Furthermore this ordering is (partially) chronological with respect to insights derived in quantum gravity, such that we can regard the results at few building blocks as a guidance how to explore the dynamics of many building blocks.

8.1 Microscopic building blocks: asymptotic expansion and the measure factor In this spirit one may ask what one can learn about discrete quantum gravity models, when one considers only one fundamental building block, e.g. one simplex. In spin foam models [21, 22] this corresponds to a single vertex amplitude, the amplitude dual to the simplex, for which one is interested in its relation to other discrete (classical) models of gravity like Regge calculus [17, 18]. A very popular tool to extract geometric information from the vertex amplitude is by considering asymptotic expansions by uniformly scaling up the geometric data, frequently SU(2) spins, to macroscopic sizes and identifying the dominant contribution in terms of an effective action. The scaling of the spins can also be understood as the limit ~ → 0, such that asymptotic limits are frequently called semi–classical. The prototype of this idea is the Ponzano–Regge model [23,80], a spin foam model of 3D quantum gravity, which has the SU(2) 6j symbol as its vertex amplitude. The famous formula by Ponzano and Regge [23], which has been derived since then by a plethora of other methods [27, 91–96, 99, 100], shows that the 6j symbol is proportional to the (cosine of the) discrete Regge gravity action associated to a tetrahedron constructed from SU(2) spins. Similar results [24–27] also exist for the modern 4D spin foam models [73–75], of which many are obtained via a coherent state approach [28, 29]. The vertex amplitudes are usually constructed via a specific contraction of invariants (under the action of the group) in the tensor product of vector spaces of irreducible representations of the group [77]. In case of the 6j symbol, this accounts for a particular contraction of four Clebsch–Gordan coefficients. One way to construct these invariants is by group averaging a tensor product of states (living in the vector space associated to the irreducible representation); for specific choices, namely the eponymous coherent states [90], this leads to non– trivial invariants. Furthermore, the coherent states are peaked on classical geometric notions, e.g. for SU(2) they are labelled by a vector ~n ∈ S 2 . In the asymptotic expansion of the vertex amplitude one scales up the representation labels associated to the simplex and evaluates the amplitude on the points of stationary phase of the group integrations. Due to the geometric data carried by the coherent states, the stationary point

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8 Discussion conditions allow for a clear geometric interpretation, e.g. in 3D, the three vectors labelling a 3– valent invariant, which is P unique in SU(2) up to normalisation, form a (coherent) triangle [81] if the vectors sum to zero, i.e. ni = 0. This is also known as the closure constraint of the triangle. i ji ~ Then these vectors are interpreted as the edge vectors of the triangle. This geometric interpretation makes it straight–forward to identify the dominating phase as the Regge action associated to the simplex, yet again as for the 6j symbol, the amplitude oscillates with the cosine of the Regge action. Despite these successes and positive indications underlining the relation of spin foam models to discrete gravity, it is generically not possible to derive the entire leading order contribution of the asymptotic expansion using the coherent state approach. This missing part, sometimes referred to as the ‘measure’, which one obtains from the determinant of the Hessian matrix, should not be underrated: Essentially it gives us a very first glance at the measure on the space of geometries chosen in spin foam models. Furthermore, it is the most directly accessible, yet naive, quantum correction. The inability to compute these measure factors, besides numerical results [26, 96], even for the well–known SU(2) 6j symbol, is rather troubling for the coherent state approach, but can be overcome by introducing modified coherent states.

8.1.1 Modified coherent states and the relation to first order Regge calculus To solve the issue illustrated above for the SU(2) 6j symbol, we introduce new ‘smeared’ coherent states in [30], see also chapter 2. We would like to emphasize that the purpose of this work is not to invent yet another method to derive the Ponzano–Regge formula [23], which is known for over 40 years, but to test the idea of new coherent states in the context of a well–examined example. These new coherent states can be understood as follows: Normally, the familiar SU(2) coherent states are eigenstates of highest weight of a generator of rotations. Thus, this state is labelled by a vector ~n ∈ S 2 , which labels the axis of rotation. Generically these states are only defined up to a phase, an issue which is not entirely understood in spin foam asymptotics and usually circumvented by choosing a canonical choice of phase [139]. For our smeared coherent states, we circumvent this issue by picking a coherent state orthogonal to the considered generator of rotations and smearing it over the circle of vectors on S 2 orthogonal to the respective generator, expressed as an integration over the smearing angle φ. In short, we construct a null eigenvector, with respect to the considered generator of rotations, defined for all representations of SU(2). The immediate advantage, which eventually allows us to compute the complete expansion (up to leading order) of the 6j symbol, is that we can split the stationary phase approximation into two steps, one with respect to the group elements and one with respect to the (artificially) added smearing angles. However, this modification comes with a caveat: due to the smearing, one generically obtains many more points of stationary phase, whose geometric interpretation is less clear and have to be suppressed. Thus, it is a priori not clear whether the new coherent states actually allow for the same geometric interpretation as the familiar ones. To clarify these two points, we first introduce modifiers for each 3–valent invariant that suppress additional stationary points. Furthermore we carefully examine the stationary phase conditions, the symmetry properties of their solutions and their geometric interpretation, which fortunately coincides with the standard coherent state construction, see e.g. [96]. This property is actually deeply rooted in the fact that the invariant subspace of the tensor product of three SU(2) representations is one–dimensional. Indeed, this agreement with the familiar coherent state approach allows us to perform the stationary phase analysis by parts, that is first with respect to the smearing angles and afterwards with respect to the group elements. This computation can actually be performed for arbitrary planar 3–valent graphs. Interestingly, after a variable transformation from group elements to (exterior) dihedral angles, the resulting effective action for the 6j symbol turns out to be the first order Regge action [97] associated to one tetrahedron1 . This already establishes 1

In first order Regge calculus, both edge lengths {l} and dihedral angles {θ} are independent variables. In order

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8.1 Microscopic building blocks: asymptotic expansion and the measure factor the connection to discrete gravity, which is essential to perform the remaining stationary phase approximation with respect to the angles: The geometric interpretation inherited from Regge calculus allows us to explicitly calculate the determinant of the Hessian matrix by employing several identities of the angle Gram matrix adopted from constantly–curved simplices [98] and rederive the well–known Ponzano–Regge formula [23]. Additionally, our formalism allows to study higher order corrections (in principle). In particular we prove a conjecture in [99,100] on the oscillatory behaviour of higher order corrections that states that sequent orders in the asymptotic expansion alternate between oscillating like a cosine and a sine, e.g. the leading order, i.e. the Ponzano–Regge formula, oscillates like a cosine, while the next–to–leading order oscillates like a sine. Moreover, we derive a recursion relation for the full 6j symbol similar to [101, 102]. For future applications of the methods developed here, the hope is that one can apply the modified coherent state idea e.g. to the SL(2, R) 6j symbol, which corresponds to a Lorentzian version of the Ponzano–Regge model, see also [27, 140]. In comparison to the SU(2) case, one certainly faces the complication of a more intricate representation theory and non–trivial invariant subspaces of the tensor product of three irreducible representations. Fortunately, the tools developed for first order Regge calculus [97] can be readily applied to the 4D spin foam model introduced by Barrett and Crane [73], such that we can derive the measure factor for the vertex amplitude of a modern spin foam model for the first time.

8.1.2 Measure factor for a 4D spin foam model Falling into the same category, the paper [31] in chapter 3 can be seen as a direct follow up of [30]: moving away from 3D, we consider the simplest, yet non–trivial, 4D spin foam model, the Barrett– Crane model [73]. Starting from a nice identity of its vertex amplitude, the 10j symbol, in [27] we restrict our attention to strictly geometric contributions to the amplitude. By ‘geometric’ we refer to two facts: first, the asymptotic expansions of most 4D spin foam models not only contain geometric sectors, but also non–geometric ones, like a BF sector. See [141] for a discussion in the EPRL–model [74]. Second, in the Barrett–Crane model, the tetrahedra along which two 4– simplices are glued together have the same areas (of triangles) yet their shapes generically do not match. Because of such issues this model is nowadays disclaimed to be a viable theory for quantum gravity. For a recent discussion on these issues and a more positive view, see [103]. If we restrict the identity of the 10j symbol from [27] to the geometric sector, it exhibits a nice interpretation as first order (area) Regge calculus [104–106], a theory in which areas and 4D dihedral angles are independent variables2 . Its equations of motion with respect to the areas (of triangles) impose local flatness, i.e. the deficit angles are required to vanish, while the non–matching of tetrahedra gives rise to metric discontinuities [105, 106]. In the asymptotic expansion of the 10j symbol however, the areas of triangles are uniformly large and fixed, and one computes the stationary phase approximation of the integration over dihedral angles: On the points of stationary phase one obtains the action of standard area Regge calculus, while the determinant of the Hessian can be straightforwardly computed by applying the methods developed in [30], inspired from curved Regge calculus [98]. Remarkably, the so–found measure factor has a nice geometric interpretation as it mostly consists of products of volumes of (sub)simplices of the 4–simplex. The only non–explicit factor appearing is a Jacobian describing the change of variables from areas to edge lengths. If we consider a peculiar triangulation, namely to arrive back at ordinary length Regge calculus, in which the dihedral angles are given as functions of the edge ˜ ˜ ij = cos(θij ) is called the angle length, the constraint det G({θ}) = 0 is imposed via a Lagrange multiplier. G ˜ ii = 1. Gram matrix. For exterior dihedral angles, as discussed in [30], one uses the convention G 2 This theory clearly differs from ordinary length Regge calculus [17, 18, 97] in 4D, in which the areas of triangles is given by the edge lengths, and should also not be confused with other formulations like area–angle Regge calculus [60, 142]

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8 Discussion two 4–simplices glued along all of their tetrahedra, giving the simplest triangulation of a 4–sphere as a product of two identical vertex amplitudes, we can absorb the implicit Jacobian into so–called glueing constraints, i.e. conditions that enforce the triangles of the shared tetrahedra to match in shape, not just in areas.

8.1.3 What is the interpretation of the asymptotic expansion? With the results derived in [30, 31], we reconfirm that the (geometric part of the) asymptotic expansion oscillates like the cosine of the Regge action and also give a first glance at the measure on the space of geometries. However, these encouraging results have to be taken with a grain of salt: even if ignoring the contributions from non–geometric sectors, the physical interpretation of the asymptotics is doubtful. A quantized simplex, blown up to a macroscopic size, asymptotically behaves like a classical simplex, i.e. a basic building block of a triangulation endowed with a discrete gravity action. Indeed, this is only the very first step in establishing the connection between spin foam models and discrete gravity, a relation that has to be generalized to larger triangulations. Unfortunately, due to the complexity of spin foam models, calculations for larger triangulations have proven to be difficult, even Pachner moves, i.e. local changes of the triangulation, have not yet been computed in the full models. In recent years doubts on the viability of the large spin limit for larger triangulations have been raised in [78]. If one scales up the boundary spins without additional assumptions on the spins in the bulk, one obtains accidental curvature constraints in the bulk forcing the deficit angles to vanish excluding a large class of Regge geometries3 . Similar constraints have also been found in [143]. Therefore it has been argued that the large spin limit by itself is not semi–classical and should be performed together with a limit, in which the number of building blocks goes to infinity, see e.g. [79]. Later on we will also argue in favour of a refinement limit, yet we do not consider spins to be suitable order parameters, since they do not represent a diffeomorphism invariant quantity, see also chapter 7 (or [88]). Instead we will also consider refinement (and coarse graining) of the boundary itself, relating the states on different boundaries by embedding maps. By imposing cylindrical consistency conditions we propose a construction scheme for the continuum theory from the discrete model. Another, at first site troubling fact is the oscillation behaviour of spin foams. Instead of oscillating like ei SR , which one would understand as a usual behaviour of a path integral, the vertex amplitude asymptotically oscillates (to leading order) like 21 (ei SR +e−i SR ) = cos(SR ). This can be understood as a sum over orientations: In the asymptotic expansion, two conjugated stationary points are contributing that correspond to two different orientations of the simplex. There exist several attempts to suppress this sum over orientations by introducing a notion of causality in spin foam models [144, 145] and it has been argued in [146] that the sum over orientations is a cause of divergences in spin foam models. On the other hand there are important reasons why one should sum over orientations: in gravity, the dynamics (and time evolution) is generated by constraints, such that time evolution itself is a gauge transformation. A path integral should therefore act as a projector onto physical states, i.e. states satisfying the constraints. To realize this property, it is heuristically necessary to integrate over positive and negative values of lapse and shift, which one can understand as evolving forwards and backwards in time or, on the discrete level, as summing over orientations. Of course, these statements are controversial and heuristic, and ignore several subtleties involved in this [20, 147–150]. Still one can also see the asymptotic expansions of spin foam models as a starting point to a different consideration. Assuming the connection between spin foam models and Regge calculus also generalizes to larger triangulations, namely also larger spin foams oscillate as the Regge action 3

Such a feature cannot be observed for asymptotic expansion of a single simplex, since there all spins are on the boundary.

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8.2 Lectures from triangulation independence: Regge calculus of the underlying triangulation does, one can ask what one can learn about spin foam models from the perspective of Regge calculus. This can indeed be insightful, e.g. it might be used to fix ambiguities in the definition of spin foam models. For example, the edge and face amplitudes in spin foam models, in the literature often referred to as the spin foam measure, are not uniquely defined and are frequently determined by kinematical considerations, e.g. invariance under subdivision of an edge or a face [76, 151, 152], which one can understand as a weak notion of diffeomorphism symmetry. These choices influence the degree of divergence of the model [52–55]. Instead we would like to advocate dynamical principles to fix the ambiguities of the models, where the most desirable, and yet most difficult to achieve, is (discrete) diffeomorphism invariance.

8.2 Lectures from triangulation independence: Regge calculus As explored in previous work, diffeomorphism invariance is generically broken by discretisations [41–43], hence also in most spin foam models4 , but one can try to resurrect it by consecutively improving the discretisation by a renormalization / coarse graining procedure. The heuristic idea is the following [43]: By infinitely refining and improving the discretisation, one reaches a fixed point, where the system is, by definition of the fixed point, triangulation independent, at which diffeomorphism symmetry is restored. On the other hand, starting with a system that is diffeomorphism invariant in the discrete, i.e. invariant under vertex translations, one can, e.g. move a vertex on top of another one; this induces triangulation independence as well. This relationship can be understood as follows: the consecutive coarse graining and thus improving of the discretisation can be interpreted as ‘pulling back’ the continuum dynamics onto the discretisation. Once achieved, the dynamics, in its entirety, is represented on the discretisation and with it the entwined symmetry. Thus it is irrelevant how fine or coarse the discretisation is, since it already captures the full dynamical information. Such discretisations are called ‘perfect discretisations’ [61, 62]: a demonstrative example is 3D (classical) gravity, whose equations of motion tell us that spacetime is locally flat. 3D Regge calculus captures this perfectly, since it is built up out of intrinsically flat tetrahedra, which are glued together in a flat way, i.e. the deficit angles vanish. Furthermore, the associated discrete Regge action is invariant under vertex translations. However, this symmetry is broken if one considers 3D gravity with a non–vanishing cosmological constant [41, 42], which describes constantly curved spacetime. Then Regge calculus with flat tetrahedra does not capture the continuum dynamics and the deficit angles (on the edges) no longer vanish. However, if one refines the triangulation, one realizes that the deficit angles decrease and the approximation is improved. In the infinite refinement limit, the symmetry gets restored. Alternatively, one can also iteratively improve the tetrahedra to capture the dynamics, which eventually leads to constantly curved building blocks described by the action for (constantly) curved Regge calculus [60]. Again the deficit angles vanish and the action is invariant under vertex translations. Certainly, the construction of such a perfect discretisation for non–topological theories is much more difficult. We will comment on this later on. The previously mentioned conjecture relating diffeomorphism symmetry and triangulation independence in [43], which also appears in a slightly different form in [88], allows us to interpret triangulation independence as a dynamical principle. Since the modern spin foam models are deliberately constructed not to be triangulation independent5 , we instead examine triangulation independence in (linearized) Regge calculus in chapter 4, see also [59], to determine whether it can be used to fix the ambiguities in the definition of the path integral, in particular the path integral 4

The role of diffeomorphisms in spin foam models has been discussed in [46], where their action has been identified as vertex translations. This concept is realized in the 3D topological models, such as the Ponzano–Regge [23, 80] or the Turaev–Viro model [124]. 5 This is partially rooted in the prejudice that any triangulation invariant theory must be topological.

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8 Discussion measure. Indeed, in this context, by requiring triangulation independence one might hope for an anomaly–free path integral measure with respect to diffeomorphisms, see also [51] for a discussion in spin foam models. In contrast to the various other proposals for measures in Regge calculus, see e.g. [107–109], this is the first one to be based on a dynamical principle and, due to the relation to spin foam models, may serve as a blueprint for possible measures in spin foam models. In this context we study triangulation independence in chapter 4, see also [59], on the local level of Pachner moves [82, 83]: Pachner moves are local changes of the triangulation, whose consecutive application can transfer a triangulation of a manifold into any other triangulation of the same manifold. Therefore it is sufficient to consider just the simplicial complex subject to the Pachner move. By linearisation we mean that the Regge action is expanded (up to quadratic order) around a flat background solution, here vanishing deficit angles. The perturbations around the background are then considered to be the new dynamical variables, whereas we regard the path integral measure to be a function of the background edge lengths. In fact, the flat background structure allows us to explicitly compute the Hessian matrix, i.e. the new ‘propagator’ of the theory, which can be straightforwardly written in an almost factorising form of geometric quantities, like volumes of simplices. The calculations themselves use and extend identities and results from [153] and [154– 156], see also [112, 157]. Furthermore, in case of the 4–1 and 5–1 move in 3D and 4D respectively, we identify the null eigenvectors associated to the vertex translation symmetry of the vertex inside the (coarse) simplex. Unsurprisingly, full triangulation independence is only achieved in 3D, where the theory is topological and the action itself is fully invariant under all Pachner moves. As a result, the path integral measure is (almost) uniquely determined in a local ansatz, which factorises with respect to the (sub)simplices of the triangulation. Remarkably, this result is compatible with the Ponzano–Regge asymptotics [23], even for the assignment of the numerical constants. No doubt, this is to a large degree due to the topological nature of 3D gravity, but it shows that triangulation independence is indeed a useful requirement to fix the ambiguities of a theory. Naturally in 4D, full triangulation independence cannot be achieved due to several reasons. The first one, derived in [59, 158], is the fact that the Regge action is not invariant under all Pachner moves, to be more precise the 3–3 Pachner move. The calculations in [59] are performed around a flat background solution, which can be guaranteed for most Pachner move, since there is at least one dynamical edge, which can be chosen such that the deficit angles vanish and the solution is flat. In the 3–3 move however, all edges are in the boundary and thus the deficit angle located at the single bulk triangle is completely determined by this boundary data. Unless these allow for a flat geometry in the bulk, even the linearized Regge action is not invariant under this Pachner move6 . Indeed, in such a situation the intrinsically flat simplices cannot capture the curvature, which as a consequence leads to a broken diffeomorphism symmetry in larger triangulations. Again, only if the boundary data allow for flat solutions in the bulk, the Regge action possesses a vertex translation invariance [44, 45]. In particular since we intend to study quantum gravity in regions with possibly large curvature, the Regge action – and presumably also spin foams – on a coarse triangulation is not a good approximation in these regions and has to be improved [41, 42]. Nevertheless, it is still instructive to discuss invariance under the two remaining Pachner moves, namely the 5–1 and 4–2 Pachner move. In both cases one can define flat background solutions and we show explicitly that the (linearized) Regge action actually is invariant under both these moves. Surprisingly, the Hessian matrix is strikingly similar to the 3D case, namely highly factorising with respect to (sub)simplices, such that one can define a measure factor analogous to the 3D case that is ‘almost’ invariant. However, despite these nice properties, full triangulation invariance of the path integral cannot be achieved due to the appearance of an overall factor in the Hessian matrix that at first glance appears to be non–factorising. 6

The two configurations in the 3–3 move differ in the triangle that is shared by all three 4–simplices. The Regge action is only invariant under this move, if the deficit angle at the single bulk triangle vanishes in both configurations.

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8.2 Lectures from triangulation independence: Regge calculus

8.2.1 Triangulation independence implies non–locality This overall factor is the main focus of the paper [84] presented in chapter 5, in which we derive its geometric interpretation: This factor vanishes if the six vertices involved in the 4D Pachner move lie on a 3–sphere [110], see also [111]. For the 3–3 and the 4–2 Pachner moves, this can only happen if the boundary data is chosen in such a way that all vertices can lie on the same 3–sphere. Yet in case of the 5–1 move, for which the classical action is invariant under translations of the inner vertex, see e.g. [59], such situations can always be constructed by moving the inner vertex onto the circumscribing sphere of the coarse simplex. This can either mean moving the vertex on top of another one, resulting in degenerate simplices, or moving it outside the coarse simplex, which corresponds to a change of relative orientation of one or more simplices [112]. Indeed, if this occurs the entire Hessian matrix vanishes and the integral diverges, which concurs with the thesis in [146] that the change of orientation, by moving the inner vertex outside the coarse simplex, is the cause of divergences in spin foam models. On the other hand we argue below that in spin foams, similar to the asymptotic expansion discussed above, one in fact has to sum over orientations in order to impose Hamiltonian and diffeomorphism constraints. More importantly, in case the vertex is outside the coarse simplex and at the same time on its circumscribing sphere, the overall factor vanishes, but none of the involved 4–simplices is degenerate, i.e. their volume is non–zero. From this fact we show that this overall factor is non–factorising, i.e. it cannot be written as a product of amplitudes associated to (sub)simplices, and it is necessarily non– local (with respect to the simplices of the triangulation), since its properties can only be inferred if the relative positions of all six vertices are known. Therefore, triangulation independence in 4D Regge calculus implies non–locality. Furthermore, it shows that the factorising measure derived for the Barrett–Crane model in chapter 3, see also [31], is not invariant under Pachner moves, yet the (non–local) change of variables from areas to edge lengths might change this statement for larger simplicial complexes. In fact, the appearance of non–local couplings is not surprising: since 4D (discrete) gravity is a theory with propagating degrees of freedom and both the 4–2 and the 5–1 Pachner move are coarse graining moves, in the sense that they decimate degrees of freedom, one expects non–localities to occur. Take for example the 2D Ising model [159]: the initial theory defined on a 2D lattice just describes nearest neighbour interactions between the spins located on the lattice sites, encoded in the Hamiltonian of the system. One way to coarse grain this is by a decimation procedure: every second Ising spin is summed over and absorbed into a new effective Hamiltonian. However, already after the first step, this new Hamiltonian describes not only nearest neighbour interactions with respect to the new lattice, but also next–to–nearest neighbour and multiple spin interactions; the theory has left the initial space of models, now containing non–local interactions. Coarse graining techniques as the one sketched above are called real space renormalization techniques [115], in contrast to momentum space renormalization. Frequently, unless one is dealing with a topological theory, these approaches are troubled with the appearance of non–local interactions enlarging the theory space, which can be naively tamed by introducing ad hoc approximations and restrictions on the allowed couplings. However, it is often not clear how good these approximations are and the non–localities effectively reduced the amount of available real space renormalization methods. One of the most prominent, yet brutal in the approximation, is the Migdal–Kadanoff scheme [113, 114] that outright removes non–local interactions, yet is still able to predict the phase transition for the Ising model qualitatively. At first sight, this seems to diminish the possibilities of improving Regge calculus or even spin foam models in this manner. This is partially due to a particular perspective of these models and the coarse graining scheme: the initial setup of the models is building an amplitude out of basic building blocks that interact locally. In spin foam models these are the vertex, edge and face amplitudes as functions of representation labels, in Regge calculus it is the additive Regge action as a functional of edge lengths. Now the coarse graining schemes, e.g. Pachner moves, intend to

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8 Discussion locally manipulate the discretisation, yet at the same time keep the basic building blocks, and the degrees of freedom describing them, as the fundamental ingredients. In particular Pachner moves, as local transformations, keep the boundary data unchanged. However, these data, while suitable to describe physics on the smaller scale, may be very inefficient to capture the dynamics at coarser scales, apparent in the appearance of non–local couplings for interacting theories. Therefore the question arises whether one can choose more appropriate degrees of freedom to avoid non–local interactions. One suggestion in Regge calculus would be to use area–angle variables [60, 142], since they are more closely related to the construction methods of spin foam models, yet it is not clear whether one can avoid non–localities in this framework. However in practice, it appears to be impossible to circumvent non–localities entirely, such that it is imperative to employ approximations. Indeed, it would be desirable to have a coarse graining scheme capable of controlling both the degree of non–locality and the quality of the approximation. To do so it is indispensable to isolate the relevant degrees of freedom from the less relevant ones, which in most cases will be recombinations of the finer degrees of freedom into coarser, collective ones. Thus we argue that a coarse graining scheme should necessarily affect the boundary data as well.

8.3 A change of perspective: Tensor network renormalization A possible solution to these issues lies in a change of perspective: instead of keeping the initial basic building blocks fixed and changing the discretisation only locally, one can employ more non– local changes, which modify the building blocks and also affect the boundary data. Essentially, the idea is to replace several locally interacting building blocks by an effective building block, which again interacts locally with the surrounding building blocks. In general, one can construct such a new building block by integrating out / summing over internal degrees of freedom, such that the new object depends on more boundary data than the initial one did. In a way, one ‘stores’ the non–localities inside the new object by paying the prize of an increasing amount of boundary data. This increase of the boundary data causes two complementary issues. The first one affects numerical realizations of such a coarse graining scheme. Consecutively performing this procedure gives an exponential growth of the boundary data, which requires the introduction of a truncation. The second one, even more severe, is the question of interpretation of the new building blocks and their comparability to the previous ones. In general, the larger boundary of the new building block is endowed with a ‘larger’ Hilbert space7 , such that it is not straightforward to compare the two amplitudes and interpret them, i.e. the essential task of any renormalization group approach. In this particular example of coarse graining, one would like to kill two birds with one stone by suitably ‘reducing’ the boundary data of the new building block, thus restoring the relation to the previous building block (and the interpretation) and at the same time truncating the boundary data to enable efficient numerical algorithms. To put it differently, one has to define embedding maps between Hilbert spaces of different size / complexity, which redefine the fine data in terms of the coarser ones and also provide an approximation. A priori these embedding maps can be chosen arbitrarily, but their choice determines the quality of the approximation and also the interpretation of the new model. The last two conditions are complementary and essentially require the embedding maps to be compatible with the dynamics of the system, i.e. the dynamics should determine the embedding maps. Therefore, the embedding maps should combine the fine degrees of freedom into coarse ones in such a way that the symmetries of the system are preserved to allow for an intuitive interpretation. Among these collective degrees of freedom, it has to identify and retain the most relevant ones, such that a good approximation can be achieved. Interestingly, although working in a very different context, condensed matter theory faces similar issues in the quest to describe many body quantum physics. Even though starting from a simple lattice system equipped with a Hamiltonian describing local interactions, the systems are generically 7

For an infinite dimensional Hilbert space, ‘size’ refers to the complexity of the discretisation.

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8.3 A change of perspective: Tensor network renormalization not solvable, yet one would like to extract the collective dynamics of many degrees of freedom via a real space renormalization method. Indeed, there exist several new methods tackling these issues, which can be summarized as providing ans¨atze for groundstates of the Hamiltonians, see e.g. matrix product states (MPS) [118, 119]. In this context, we are particularly interested in tensor network renormalization [85, 86], since it is a proper realization of the ideas illustrated above. A tensor network can be understood as a reformulation of the lattice system: The local degrees of freedom on the lattice are locally encoded into a tensor Tabcd... , where the rank of the tensor encodes the valency of the vertices of the lattice. The partition function Z of the system is then computed by contracting all tensors X Z= Tabcd Tb0 c0 ad0 . . . (8.3.1) a,b,c,d,...

according to the combinatorics of the network, here for 4–valent tensors. Thus, the tensors T encode the dynamics of the system. Yet the task to compute the partition function Z is still as difficult as in the original setup, such that one would like to introduce a coarse graining scheme that can approximate Z by a contraction of a coarser tensor network made up out of effective tensors T 0 , which possess a maximum index range χ8 . To do so, tensor network algorithms aim to iteratively combine a certain amount of tensors into a new tensor, which naively suffers from an exponentially growing index range. Furthermore, since the tensor contains the dynamical information of the system, the new tensor should also be comparable to the previous one in order to observe the change of dynamics at different scales. In fact there exist many different implementations of tensor network renormalization differing in the way how they construct the embedding maps, relating fine to coarse boundary data, from the tensors, yet all of these algorithms employ singular value decompositions, in short SVD. To do so one rewrites a collection of tensors as a (possibly asymmetric) matrix by an appropriate grouping of tensor indices, to which the SVD can be applied. Then, the SVD can be understood as redefining the variables and ordering them according to their relevance, which can be readily read off from the size of the associated singular value. This allows us to employ a truncation scheme, e.g. by keeping only the χ most relevant degrees of freedom, i.e. the χ largest singular values. The new effective tensor T 0 , labelled by the new degrees of freedom, is the starting point for the next coarse graining step. Thus the idea is to use this particular approach as a coarse graining scheme for spin foam models. Interestingly, spin foams fit quite naturally into the language of tensor networks as it has been described in [37]: the projectors onto invariant subspaces associated to the edges, can be directly written as a tensor of the same rank as the valency of the edge, i.e. the number of faces sharing the edge. Similarly, one can also absorb the face amplitudes into the tensor. A complication exists however for spin foams, as well as for lattice gauge theories [37, 121]: one has to ensure that each face carries the same representation label and thus, also all tensors on the edges sharing this face must carry the same label in their respective index. Therefore one has to introduce auxiliary tensors on the faces that fix the labels of the adjacent dynamical tensors to be identical. Yet, even though spin foam models exhibit a nice translation into tensor networks, one faces two obstacles if one intends to apply tensor network renormalization to them: • Most tensor network renormalization algorithms are defined for spin systems defined on a 1–complex, e.g. a 2D lattice. By spin system we mean a system with a global symmetry, like the Ising model (without external magnetic field), instead of a local gauge symmetry, like Z2 lattice gauge theory. On the lattice, these systems can be modelled as a vertex model, e.g. with the tensors of the network associated to the vertices. Here no auxiliary tensor are required on the faces, such that the whole network only consists of one type of tensors. Thus, 8

This χ is frequently called the bond dimension.

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8 Discussion in order to apply these methods to spin foams, one has to generalize the algorithms to higher dimensional complexes and modify them to deal with the more complicated networks. • The second, equally troubling issue lies in the underlying symmetry group of spin foam models. Whether one examines Riemannian or Lorentzian gravity, one picks either SO(4) or SL(2, C) as the underlying symmetry groups. These groups generically come with infinitely many representation labels, which would result into tensors with equally ranging indices. At first sight, this appears to doom feasible numerical simulations. Indeed, each of this issues is very difficult to tackle in full generality: the first one requires the development of a new coarse graining scheme within the tensor network formalism (currently work in progress), while the implementation of the full symmetry group leads to infinities in the partition function. A solution to circumvent this issue is unknown to the author of this thesis. Thus, instead of tackling these issues head on, it is better to come up with suitable approximations that make currently available methods applicable, while still learning something new about spin foams in the process, e.g. whether tensor network renormalization is (or can be) a useful tool to study the collective dynamics of spin foams. This idea lead to the development of analogue spin foam models, called spin nets [37–39], which can be directly understood as a dimensional reduction of spin foams: the model is defined on a 1–complex, i.e. a graph, e.g. a 2D (regular) lattice, where the dynamical ingredients of spin foams, face weights and projectors (on the edges), are associated to lower dimensional objects, i.e. weights on the edges and projectors on the vertices. Such systems can be readily translated into tensor networks and tensor network renormalization is applicable in principle. Yet due to the infinite number of representation labels one has to sum over for the Lie groups of interest, the index range of the tensors is infinite as well, which poses a serious challenge for numerical simulations. This can be circumvented by replacing the Lie group by a finite group [37–39], which naturally come with only finitely many representation labels. See also [160] for a definition of spin nets and foams for finite groups. Another option are quantum groups like SU(2)k [87], see also chapter 6, which are equipped with a cut–off on the representation labels depending on the level k [122, 123]. The dimensional reduction might appear ad hoc and is mainly motivated by the restrictions of the coarse graining method, which would be dishonest to neglect. Therefore, we would like to motivate further, why we think even these simplified models can teach us something about the many–body behaviour of spin foams. Actually, it is known that 2D lattice gauge theories are equivalent to 1D spin systems, and also 4D lattice gauge theories have several properties in common with 2D spin systems [161], e.g. one noticed similarities between the 2D Ising model and the 4D Z2 gauge theory. Similarly, we would like to interpret the results for spin nets as indications for spin foams. Furthermore, as we also show in [87] (see also chapter 6), spin nets can be interpreted as highly anisotropic spin foams, taking the form of a ‘melon’, similar to [54]: Such a spin foam consists of two vertices connected by many 4–valent edges, which are basically dual to tetrahedra. Then coarse graining of spin nets can be interpreted as defining new effective tetrahedra, which might actually also appear in simplicial 4D spin foam models. Certainly, this does not imply that the same fixed points emerging from spin net simulations also occur in spin foam models, yet this is supportive nonetheless. Before discussing the results of this approach, we would like to emphasize, what we are looking for and why: As thoroughly discussed above, the intent is to extract effective dynamics of many building blocks from spin net models via tensor network renormalization. Therefore we start the simulation with an initial tensor that encodes the full dynamics of the model. Under coarse graining, we approximate the partition function of a regular, square 2D tensor network by coarse graining the tensors into new effective ones as prescribed above. The tensors will generically change under this procedure until they eventually reach a fixed point, i.e. the tensors remain unchanged under further coarse graining steps. This final tensor resembles the refinement limit of the specific initial

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8.3 A change of perspective: Tensor network renormalization model and encodes its continuum dynamics. Interestingly, these different continuum dynamics can be distinguished by their excited singular values.

8.3.1 Coarse graining spin nets: finite groups In order to demonstrate the progress in coarse graining spin net models over the past three years, let us briefly summarize previous results. The very first work [37, 38] is based on Abelian finite groups Zn , and asks the question, in the spirit of related work [41–43], whether broken symmetries get restored under coarse graining. In this context, not (discrete) diffeomorphism symmetry (or reparametrization invariance) is of interest, but rather the global BF symmetry of the system, which is also called the ordered phase. This phase is parametrized by assigning exactly a constant weight to all representation labels; in the dual picture for the Ising model, this corresponds to the state in which all spins are equal. Since this state already defines a fixed point of the renormalization scheme, one has to break this symmetry, e.g. by introducing a cut–off on the representation labels, and study the flow of these ‘Abelian cut–off’ models under coarse graining. Indeed, [37] already gives a glance at the power of tensor network renormalization, by providing the first phase diagrams for spin foam related models. Depending on the initially chosen cut–off on the representation labels, the models mainly flow to the (analogue) BF phase or to the degenerate phase, in which only the trivial representation is excited. The latter phase is also called disordered or high–temperature phase. The next logical step is to extend this approach towards non–Abelian finite groups, the simplest of which is S3 , the permutation group of three elements, in [39]. In fact, this work is remarkable in several ways: it is the first study of the effective dynamics of an (analogue) spin foam model that possesses similar features to modern spin foam models [73–75], namely a notion of simplicity constraints [66], which are implemented by special functions analogous to the holonomy formulation of spin foams in [69]. In fact, these simplicity constraints get encoded into the projectors associated to the vertices, which, in comparison to lattice gauge theories, project onto a smaller invariant subspace (of the tensor product of irreducible representations meeting at the vertex). Naturally, the question arises, whether the simplicity constraints persist under coarse graining or whether the model flows back to the standard lattice gauge theory phases, such as the BF or the degenerate phase9 . In order to answer this question, the coarse graining mechanism has to be adapted in such a way that it preserves the symmetries of the model, in this particular case the group symmetry encoded in the tensors. Therefore a symmetry protecting algorithm has been invented in [39] that works as follows: before performing the SVD, the system is rewritten into the so–called ‘recoupling basis’, a transformation in the representation theory of the group that brings the tensor into a block diagonal form10 , where each block is then labelled by a pair of representations (ρ, ρ0 ). Then the SVD is performed for each of the blocks (ρ, ρ0 ), which is not only computationally more efficient, but also preserves the interpretation of the variables: The representation labels of the blocks, called intertwiner channels, are the new labels of the effective tensor. The work [39] uncovered several remarkable results. First of all, one can determine the phase of the system from the excited intertwiner channels and, moreover, determine directly whether the system is in a standard lattice gauge theory phase. If it is not, one will observe excitations in channels (ρ, ρ0 ), where ρ0 6= ρ∗ . This is not only a generalization of the initial model, which started with (ρ, ρ∗ ), but sheds a new light on the interpretation of spin foams: for spin foams the excited intertwiner channels are the relevant degrees of freedom, specifying the model and the 9

In lattice gauge theory, BF theory is also known as the weak coupling limit, whereas the degenerate phase is also known as the strong coupling limit. In condensed matter, one would refer to these phases, in analogy to the Ising model, as the low temperature or ordered phase or the high temperature, disordered phase respectively. 10 In fact, [37] already possessed a similar scheme in its algorithm, by preserving the Gauß constraints (at each vertex) under coarse graining. Similarly this protected the interpretation of the labels in the edges in terms of representation labels, yet for a much simpler system.

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8 Discussion dynamics. Additionally, as soon as channels with ρ0 6= ρ∗ are excited, one can interpret this as an implementation of simplicity constraints. Despite these interesting insights, the results [39] are mixed: On the one hand one has been able to find one non–trivial fixed point, i.e. with channels (ρ, ρ0 ) and ρ0 6= ρ∗ excited, yet in order to flow to it required a high level of fine–tuning and rather low accuracy of the numerical algorithm. As soon as a higher accuracy of the algorithm has been used, i.e. more degrees of freedom were stored in each coarse graining step, the system either flowed to the BF phase, the degenerate phase or BF theory on a normal subgroup, i.e. (analogue) phases known from lattice gauge theories. Thus, even though additional structure in form of the new fixed point has been found, the phase diagram only showed extended phases for the standard lattice gauge theory phases.

8.3.2 Quantum group spin nets Keeping the results of [39] in mind, the paper [87] presented in chapter 6 extends them in a seminal way. The first, non–trivial change is the definition of spin net models on the quantum group SU(2)k [122, 123], which raises these models to a higher level of plausibility: instead of considering finite groups, which are difficult to relate to gravity, quantum groups are expected to model gravity with a non–vanishing cosmological constant. The heuristic argument is that the level k of the quantum group determining the cut–off on the representation labels, here the maximal allowed spin, is anti–proportional to the size of the cosmological constant. Indeed, the very first spin foam model defined on quantum groups is the Turaev–Viro model [124], which describes 3D gravity with a cosmological constant11 . In recent years, also the modern 4D spin foam models have been defined on quantum groups [130–132] and the possibility to link (2+1)D loop quantum gravity to the Turaev–Viro model has been explored as well [125–129]. Indeed, SU(2)k combines the advantages of providing a natural cut–off onto the representation labels with a much more realistic applicability to physics. Furthermore, in contrast to finite groups, quantum groups for different levels k almost have the same representation theory, which makes the study of ‘larger’ quantum groups straightforward (neglecting the increase of computational cost). Of course, quantum groups, which are quasi–triangular Hopf–algebras [122], are much more complicated objects than groups. These complications force us to use a different initial parametrization in comparison to [39]. There, simplicity constraint enforcing functions have been used to parametrize the initial tensor, which in the quantum group models however lead to a violation of the quantum group symmetries. Instead we choose initial data lifted from so–called ‘intertwiner models’ defined in [162], which can be understood as simpler versions of spin nets, since their edges carry only one irreducible representation and thus a simpler Hilbert space. Remarkably, by requiring triangulation independence and certain restrictions on the allowed representations, one can construct a plethora of ‘fixed–point intertwiners’ of these models, i.e. topological theories, which we use as a new parametrization for our quantum group spin nets12 . In order to lift these intertwiners to the full spin nets, we revisit Reisenberger’s principle [163], a concept prescribing how to uniquely construct higher valent intertwiners from three–valent ones. Additionally, in order to define the dual representation we invent a graphical calculus, which considerably simplifies the calculations and allows us to adapt the symmetry preserving algorithm, and thus the notion of intertwiner channels, invented in [39] to quantum groups. Indeed, as it turns out, this new parametrization, which also permits us to study linear combinations of initial intertwiners, reveals a very rich structure: The first remarkable difference with respect to [39] is that for each different choice of initial intertwiner (without superposing it with other intertwiners), the system flows (under coarse graining) to a different fixed point, in fact one discovers a whole family of non–trivial fixed points, which 11

In fact, it is closely related to the Ponzano–Regge model, with the difference that the vertex amplitude is the q–deformed 6j symbol instead of the standard SU(2) 6j symbol. In discrete, classical gravity this corresponds to replacing flat tetrahedra by constantly curved ones [42, 60]. 12 Ironically, the work [162] has been motivated by the single non–trivial fixed point in [39].

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8.3 A change of perspective: Tensor network renormalization form an interesting pattern for different levels k of the quantum group. More importantly, this behaviour is stable for high accuracy of the simulations, i.e. high number of stored degrees of freedom, and requires no fine–tuning13 . In fact, most of the fixed points are non–trivial, i.e. they have excited intertwiner channels (j, j 0 ), where j 0 6= j ∗ . Even more interesting, if one considers linear combinations of the initial intertwiners, one finds extended phases (in the parameter space) for all fixed points with phase transitions. So, what is the meaning of the fixed points? As it turns out, all of the fixed points, also the non–trivial ones, describe topological and thus fully triangulation invariant14 theories. This can actually be deduced from the fact that they can be described by finitely many, locally interacting degrees of freedom, which are all equipped with a singular value equal to one. Of the non–trivial fixed points, most fall into the category of factorising fixed points; the two representations (j, j 0 ) assigned to each edge completely decouple. In the ‘melonic spin foam’ interpretation of spin nets discussed above, this actually implies a complete decoupling of the two spin foam vertices, a rather pessimistic scenario, should these fixed points also occur for the full theory. On the other hand, one finds (analogue) BF theory, which could be understood as maximally ‘glueing’ the two vertices together. Thus one is tempted to conclude that the factorizing case occurs because the simplicity constraints are too strongly imposed, leading to a complete decoupling of the basic building blocks, whereas introducing them too weakly, the system inevitably returns to analogue BF theory. Interestingly, there exist an intermediate fixed point, called ‘mixed’, which is not factorising, but also not BF . To better understand this interplay is an interesting prospect for future research. The observing reader will certainly object that these fixed points describe topological theories and wonder how one might get propagating degrees of freedom back. A preliminary way to ‘escape’ topological theories is by tuning the system towards a phase transition, where one makes the following observations: The very first peculiarity that attracts attention is that the simulations take considerably longer, the closer the system is tuned towards the phase transition. If one plots for example the singular values against the number of iterations, see e.g. in [164], one notices indications of scale invariance: For many iterations the singular values remain almost constant before eventually converging towards their final value. Actually, the closer the system is tuned towards the transition the longer it takes the system to ‘decide’ where to flow to. This behaviour can also be spotted at the phase transition of the Ising model (in a tensor network formulation), which is known to be of second order. Another observation worth discussing is the overall list of singular values, concerning the viability of the truncation: in many situations, the singular values drop off reasonably quick, such that one can truncate at a singular value, which is not only much smaller than the largest one, but also considerably smaller than the next larger one. Close to the phase transition however, the choice where to cut off is much more intricate, since the singular values drop off much more slowly, such that one may truncate degrees of freedom that, while being much smaller than the largest one, are of similar size as the smallest considered one. In short, in these cases it would be best not to truncate any degrees of freedom, which indicates that the phase transition probably is of second order and the continuum theory ‘on’ the phase transition possesses propagating degrees of freedom. In short, [87] provides a lot of positive implications for spin foams, in particular the vast amount of additional fixed points with extended phases, where the phase transitions appear to provide a possible door towards a theory with propagating degrees of freedom and implemented simplicity constraints. Yet, so far we have only glimpsed at the role of the embedding maps described above, which mainly appear in this context as providing a truncation. In the next section, we will stress the importance of embedding maps, relating coarser to finer boundary data, for the consistent 13

Of course, the term fine–tuning has to be taken with a grain of salt, since it is highly dependent on the employed parametrization. Nevertheless, it is insightful, since it indicates that the parametrization in terms of intertwiners is more suitable for spin foams. 14 In order to check this statement, the 4–valent tensors have to be split into 3–valent ones, which is part of the particular coarse graining procedure [86].

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8 Discussion construction of a continuum theory from the discrete one.

8.4 Embedding maps as time evolution In this discussion so far, one might get the impression that the main purpose of the dynamical embedding maps, e.g. obtained from tensor network renormalization, is to provide a convenient and good approximation of the dynamics, from which one can obtain an effective macroscopic description of the microscopically defined system. Indeed, as we argue in [88] presented in chapter 7, this viewpoint is too narrow and can be dramatically generalized. To do so, let us consider a spin foam giving the transition between two boundary states, each represented on a discretisation. On the one hand, this spin foam can be interpreted as one particular time evolution of the initial state to the final state. On the other hand, the two boundary discretisations can be different, e.g. the final state can be finer or coarser than the initial one, such that the spin foam provides an embedding of the initial Hilbert space into the final one. Furthermore, since spin foams implement a dynamics, this embedding is indeed dynamical. Therefore, we argue that dynamical embedding maps can be interpreted as time evolution maps. To make this idea more clear and concrete, let us elaborate on its origin: the main inspiration comes from work on the canonical formulation of classical Regge calculus [133, 134], or more general, the definition of a canonical formalism for phase spaces of different dimension at different time steps [135]. See also [136, 137] for an extension of the formalism to quantum systems. Imagine a triangulated hypersurface, for concreteness in two dimensions. This hypersurface can be locally evolved in time by glueing a tetrahedron onto it, which results in a new hypersurface at the next time step. From the perspective of the 2D hypersurface, this resembles a 2D Pachner move, depending of how the tetrahedron is glued to the initial hypersurface, either a 2–2 or a 1–3 move. The inverse moves can be understood by removing a tetrahedron (glueing a tetrahedron with opposite orientation). Remarkably, even for non–gravitational systems, this formalism always results in the appearance of constraints, namely so–called pre– and post–constraints. Pre–constraints are basically conditions that have to be fulfilled for time evolution to take place, whereas post– constraints are automatically fulfilled once time evolution has taken place. For a concrete example, consider a refining move, e.g. a 1–3 move in Regge calculus: The final phase space contains three more edges, i.e. three more degrees of freedom. To accommodate them, one has to artificially enlarge the initial phase space to describe this evolution move. As a consequence one obtains three post–constraints, essentially restricting the fictitiously added degrees of freedom. Furthermore, in the gravitational context, these post–constraints implement both diffeomorphism and Hamiltonian constraints [134, 165], where the associated gauge symmetry is interpreted as vertex translations. To put it differently, at first sight the new hypersurface has three more degrees of freedom, which however turn out to only be gauge degrees of freedom15 . For a quantum system, we argue that something similar happens if one time evolves from a coarser to a finer state. The newly added degrees of freedom are either gauge degrees of freedom or added in a vacuum state. Therefore, similar to the classical case, the refined state does not contain more information than the initial one, but represents the same information on a finer discretisation, possibly arranged in a different way. However, the issue of uniqueness forces itself into the discussion. Given one initial state on a particular discretisation, there exist many different ways to (locally) time evolve this state towards the same final discretisation, where it is a priori not clear that these different embeddings give the same final state. Therefore, one has to discuss consistency 15

From this insight, one can already deduce that refining a hypersurface by purely refining Pachner moves, i.e. 1–(d + 1) moves in (d + 1) dimensions is insufficient. Such a refining procedure leads to geometries known as stacked spheres, which are not dynamical and do not allow curvature. These stacked spheres appear e.g. in the weak coupling phase of simplicial gravity in dynamical triangulations [166] or in the melonic phase in coloured tensor models [167]. On the other hand, one can interpret them to resemble degenerate geometries with a lower Hausdorff dimension [168].

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8.4 Embedding maps as time evolution conditions for the embedding maps, which naturally translate into consistency conditions for time evolution.

8.4.1 Cylindrical consistency and the physical vacuum It is straightforward to outline the immediate conflict between an ambiguous (local) time evolution and time reversibility: assume a particular time evolution of an initial state to a final one. Then one can time evolve the final state back in a different way, which results in the same initial discretisation, yet a different initial state. To avoid this troubling property, one has to require path independence under time evolution: no matter how one chooses to locally time evolve a state to a particular final discretisation, the final state represented on this discretisation is always the same. Invoking the interpretation of time evolution as embedding maps, this leads to the following consistency conditions the embedding maps have to satisfy: Consider three boundaries b, b0 and b00 satisfying the relation b ≺ b0 ≺ b00 , i.e. b0 is a refinement of b and b00 is a refinement of both b and b0 . Then the embedding map ιbb00 should not depend on the intermediate boundary b0 , for any choice of b0 . In other words, first evolving from b to b0 and then from b0 to b00 should give the same embedding map as directly evolving from b to b00 , i.e. ιbb00 = ιb0 b00 ◦ ιbb0 . Indeed, this is an essential feature, because it allows the unambiguous comparison of two states, for concreteness ψb and φb0 , which are represented on different boundaries b and b0 . If these two boundaries have a common refinement, e.g. b00 with b ≺ b00 and b0 ≺ b00 , then both states can be embedded into the Hilbert space Hb00 and compared. If ιbb00 (ψb ) = ιb0 b00 (φb0 ), then one identifies ψb ∼ φb0 and defines an equivalence class for these states. The continuum Hilbert space can then be defined via an inductive limit, i.e. as theSdisjoint union of all boundary Hilbert spaces Hb modulo the equivalence relation ∼, in short H = b Hb / ∼. In fact, by embedding the states into the continuum (via the inductive limit), one can represent a state on a given discretisation in the continuum Hilbert space, such that the (equivalence classes of) states only implicitly depend on the discretisation they are represented on. These consistency conditions are called cylindrical consistency conditions, a concept also used in loop quantum gravity in order to relate states, more precisely spin network states, a convenient basis of the Hilbert space of loop quantum gravity [20], defined on different graphs. A spin network defined on a graph γ can be embedded into a graph γ 0 , if γ is a subset of γ 0 , denoted as γ ≺ γ 0 , by setting the SU(2) spins on all edges added to γ to obtain γ 0 equal to the trivial representation, j = 0. No restrictions on the valency of the vertices of the graph are imposed. Following our previous line of argumentation, the new degrees of freedom available in the larger graph, are added in the vacuum state, here the Ashtekar–Lewandowski vacuum [34, 35], which represents degenerate geometries. Imposing several conditions, e.g. a certain representation of the holonomy–flux algebra of loop quantum gravity and (spatial) diffeomorphism invariance of the vacuum, the F/LOST theorem [169, 170] proves that this representation of loop quantum gravity is unique. However, there is a key difference between our proposal in chapter 7 [88] and the loop quantum gravity construction: the latter is based on the kinematical states and thus employs kinematical embedding maps to relate these states across different boundaries, while we intend to use dynamical embedding maps to compute ‘transition amplitudes’ between (kinematical) states, where dynamical means that the embedding maps are supposed to impose the constraints of the theory. Let us elaborate on this further. Formulating general relativity in a canonical formalism reveals it to be totally constrained: Due to diffeomorphism symmetry of the Einstein–Hilbert action, the solutions of the equations of motion, given initial conditions, are not unique, but one can always obtain a new solution by applying a diffeomorphism. In the canonical formulation this leads to the appearance of constraints, essentially restrictions on the phase space, since one cannot solve the first time derivatives of the configuration variables in terms of their canonical momenta. These constraints are the generators of infinitesimal gauge transformations and form an algebra, in gravity this is known as Dirac’s hy-

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8 Discussion persurface deformation algebra [171]. For totally constrained systems like gravity, the Hamiltonian itself is a linear combination of constraints and is thus forced to vanish. This implies that time evolution itself is a gauge transformation. There exist several different methods to canonically quantize constrained systems in principle, where we will here focus on the quantization method by Dirac [171], see also [172] for a nice explanation, which is also employed in loop quantum gravity. Essentially the idea is to quantize the unconstrained system, i.e. the system before constraints are imposed. The associated Hilbert space is called kinematical and the challenge is to define constraint operators on this space, that realize a quantum version of the Poisson algebra of constraints [173]. A physical state is then defined to be annihilated by all constraint operators. In this context, time evolution governed by the exponential of the constraints implies that the physical state should not change, since a physical state, by definition, is annihilated by all generators of gauge transformations, i.e. the constraints. Thus, heuristically, time evolution in quantum gravity should act as a projector onto the physical Hilbert space, leaving physical states unchanged. This brief, sketchy recollection of the canonical quantization procedure of systems with constraints reveals the crucial difference between the kinematical embedding maps of loop quantum gravity and the dynamical embedding maps we propose in [88]. The kinematical embedding maps relate the kinematical spin network states, defined on a graph, to spin network states defined on a different graph. In the inductive / projective limit [34,35], these embedding maps allow for the construction of the continuum kinematical Hilbert space with a unique kinematical vacuum [169, 170], but contain no information on the dynamics. Following our arguments from the previous paragraph, a dynamical embedding map on the other hand, responsible to generate time evolution of the discrete system, must act as a projector onto the physical Hilbert space and therefore impose the (Hamiltonian and diffeomorphism) constraints of the theory. Remarkably, this heuristic picture is in nice agreement with time evolution in canonical quantum gravity: If the (dynamical) embedding maps satisfy cylindrical consistency conditions and one identifies physical states on different boundary discretisations, then the time evolved physical state remains in the same equivalence class as the original one. In fact both physical states describe the same physical situation, such that the physical state of the system is not changed under time evolution, but merely represented on a different discretisation. Similar to the kinematical construction, the dynamical embedding maps give rise to a notion of a physical vacuum; the unique Hartle–Hawking vacuum [174] of the theory. Consider an ‘evolution from nothing’, namely starting from an ‘empty’ universe encoded in the one–dimensional Hilbert space H0 = C. Time evolving this state by adding simplices, i.e. embedding it into the larger Hilbert spaces associated to the larger boundary, represents this vacuum on finer and finer discretisations up to the continuum. Yet by definition, the state remains in the same equivalence class as the unique initial one, thus uniquely representing the physical vacuum of the theory, which can be represented on any discretisation and the continuum. Indeed, this construction principle introduced in [88] has been successfully used to construct a new representation of (2+1)–dimensional loop quantum gravity [175] based on the physical vacuum (in (2 + 1)D): Instead of considering arbitrary graphs and embeddings of those into finer graphs, one restricts the boundaries to be dual to triangulations, which restricts the vertices of the graph to be 3–valent. These dual triangulations are then refined by Pachner moves instead of (in principle) arbitrary refinements. Then the uniqueness theorem of the loop quantum gravity [169,170] vacuum is circumvented by constructing a different holonomy–flux algebra, which is cylindrically consistent with respect to the new refinement procedure. In a way, this new vacuum can be seen as the dual version of the Ashtekar–Lewandowski vacuum [34, 35], since it is peaked on flat closed holonomies, thus also called the BF vacuum. Indeed, this is the physical vacuum of the 3D gravity, since it describes locally flat geometries. Note that this BF vacuum is not the first attempt to construct a vacuum for loop quantum gravity, which is not peaked on completely degenerate geometries, see

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8.4 Embedding maps as time evolution also [176] for a review on alternate representations of loop quantum gravity with non–vanishing vacuum expectation values of geometric operators, which are more suited as vacuum states for effective theories. Nevertheless, a serious word of caution is due: the dynamical cylindrical consistency conditions, and equivalently the path independence of time evolution, are highly non–trivial requirements to satisfy. Indeed, it is very unlikely to readily write down a (family of) embedding maps that can (perfectly) impose Hamiltonian and diffeomorphism constraints, since, very similar to the construction of perfect discretizations [41–43], it requires to solve the dynamics. In fact, as we also describe in [88], the examples in which it is realized all resemble topological theories including the new vacuum for (2 + 1)D loop quantum gravity [175] or the (non–trivial) fixed points obtained from coarse graining quantum group spin nets [87], see chapter 6. However, we would like to argue that achieving cylindrical consistency might not be necessary in any situation, e.g. consider a kinematical spin network state as the initial state. This state can be time evolved by glueing spin foam amplitudes, dual to simplices, to it, which should at least approximately impose diffeomorphism and Hamiltonian constraints, similar to the description in the classical case [133, 134]. If one stops after a finite number of time evolution steps, this may already suffice to compute good approximations to expectation values of sufficiently coarse grained observables, that is observables sensitive to geometric degrees of freedom on the scale of the discretisation. On the other hand, once sufficiently fine grained, the spin foam should behave as a proper projector onto the physical Hilbert space16 , i.e. in the refinement limit. To put it differently, the spin foam amplitudes have to be improved in order to act as proper projectors onto the physical Hilbert space thus closing the circle back to perfect discretisations [42, 43] and also tensor network renormalization [39, 85–87]. See also [177] for a definition of the transfer matrix from spin foams as a sum over all 2–complexes.

8.4.2 Constructing quantum space time After these very conceptual considerations, we would like to emphasize more, how one can tackle the construction of a consistent continuum theory of gravity in practise, e.g. via the previously described tensor network renormalization. Therefore, it is instructive to explain in more detail, how these coarse graining methods fit into the scheme of cylindrical consistency and how this affects their interpretation. To be more concrete, we will focus on the tensor network formulation, which also includes spin foams. Consider a single tensor for now, which we interpret to be dual to a ‘chunk’ of spacetime. Analogous to our ideas expressed above, the boundary of this chunk carries boundary data, more precisely a state in the Hilbert space associated to that boundary. The legs of the dual tensor pierce the boundary of this chunk and are thus labelled by this data, actually expressing the dependency of the tensor on this data. To put it differently, the tensor is a map from the Hilbert space associated to its boundary into the complex numbers, thus assigning an amplitude to this ‘piece of quantum spacetime’. This interpretation is fundamentally inspired by the general boundary formulation [178], a formalism invented to describe quantum mechanics, quantum field theory and quantum gravity in general regions of spacetime by assigning Hilbert spaces to the boundaries of these regions and amplitude maps to the regions itself. Given this interpretation, one might be tempted to tentatively call the tensor, or alternatively the underlying spin foam, an ‘atom of spacetime’17 . 16

Another remark on unitarity: The time evolution depicted here is not unitary, since under time evolution, kinematical degrees of freedom get projected out and cannot be recovered by time evolving backwards. Instead, one would time evolve back to the physical state ‘hidden’ in the kinematical initial state. 17 The notion of ‘atom’ must be taken with a pinch of salt: Since we are discussing background independent approaches to quantum gravity, there exists no background scale to compare this atom to. Hence, it is also fuzzy to talk about ‘larger’ and ‘smaller’ regions; only relational statements are well–defined.

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8 Discussion Of course, a tensor network does not only consist of one tensor, but of a collection of many connected ones, forming a ‘large’ region of spacetime built up by many ‘small’ building blocks. This network then gives a fine grained amplitude map by summing over the bulk degrees of freedom and keeping the fine grained boundary data, where so far no approximation has been employed. While this finer amplitude map contains the entire dynamical information, it is also defined on a larger Hilbert space than the previous one. In order to relate this new amplitude map back to the original one and thus define a renormalization group flow for the tensor / the amplitude map, the boundary cannot be left unaffected: one has to define an embedding map, embedding (or rather blocking) the highly fine grained state back into a Hilbert space assigned to a coarser boundary18 . Thus, we can compare the amplitude maps, i.e. the dynamics, before and after the coarse graining, which one could interpret as different (relative) ‘scales’. Tensor network renormalization fits nicely into these ideas, since it provides a clear method to compute the embedding maps: the singular value decomposition applied to the tensor(s), i.e. the ingredients carrying the dynamics of the system, rearranges the degrees of freedom into an orthogonal basis, ordered in their significance by the size of their associated singular value. Therefore, the embedding maps here are essentially convenient variable redefinitions together with a cut–off, the bond dimension, truncating the less important degrees of freedom. Therefore, we would like to stress again the crucial difference between the embedding maps obtained via tensor network renormalization (TNR) and e.g. the kinematical embedding maps in loop quantum gravity. In TNR the embedding maps themselves are computed from the amplitude maps, i.e. the tensors. Since these amplitude maps define the dynamics of the system, this implies that the so obtained embedding maps are constructed consistent with the dynamics, here explicitly explaining how the effective degrees of freedom on a ‘larger’ scale arise from the degrees of freedom on a ‘smaller’ scale. As a consequence, the non–triviality of the cylindrical consistency conditions for dynamical embedding maps becomes apparent in chapter 7 [88]: on the one hand, one has to determine the embedding maps consistent with the dynamics, i.e. the amplitude maps, relating the (partially ordered) boundary Hilbert spaces up to the continuum. Yet on the other hand, one requires the amplitude maps to be cylindrically consistent, i.e. it should not matter whether one evaluates a state on the coarse boundary with the coarse amplitude map or whether one embeds the coarse state into a finer boundary (Hilbert space) and then evaluates it with the finer amplitude map. The vital (and very difficult) feature here is that both embedding and amplitude maps have to be mutually consistent with one another. As already emphasized above, these consistency conditions are very difficult to realize, unless one has solved the dynamics of the continuum theory already and can ‘pull it back’ onto the discretisation. Indeed, this is a challenge very similar to perfect discretisations [41–43], which aim at constructing a discretisation without breaking diffeomorphism symmetry. In both cases, the idea is to start with a theory, here a particular choice of amplitude maps given by a spin foam model, that is generically not cylindrically consistent, but can be iteratively improved via coarse graining techniques. Concretely, one constructs improved amplitude maps by combining finer ones and dynamically embedding / blocking them back into the previous boundary Hilbert space. Hence, one may wonder whether one can find positive indications that cylindrical consistency can be achieved, e.g. via TNR in the results of [87]. Similar to other coarse graining approaches, e.g. in [43], we observe that the renormalization group flow converges and ceases in a fixed point, where fixed point means that the tensors, the amplitude maps, are not changed under the coarse graining procedure. Indeed, since in TNR the embedding maps are directly computed from the amplitude maps, on the fixed points both the amplitude and the embedding maps do not change under further 18

It may turn out that the previous / initial Hilbert space is just a subspace of the ‘full’ Hilbert space assigned to that boundary. This actually occurs e.g. in the quantum group spin nets [87]: The initial model has the restriction that only channels (j, j ∗ ) are excited, while under coarse graining (depending on the initial data) also channels (j, j 0 ) with j 0 6= j ∗ get (and stay) excited. Just restricting to channels (j, j ∗ ) under coarse graining would be too limited. In principle, one can also permit more general initial data with channels (j, j 0 ) excited.

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8.4 Embedding maps as time evolution coarse graining transformations, thus they perfectly satisfy the cylindrical consistency conditions. Then, if we interpret the embedding maps again as time evolution maps, they also realize the path independence condition and act as projectors onto the ‘fixed point theory’. Actually, these features are not surprising, since the found (non–trivial) fixed points in [87] all resemble topological field theories. This fact of the fixed points can be read off from the singular value decomposition: In the specific tensor network algorithm used in [87], the singular value decomposition is employed to split the 4–valent tensors into two 3–valent ones; the SVD itself can then be understood as a variable redefinition performed in a symmetry preserving way, retaining an interpretation of the new variables as SU(2)k representations. On the fixed points of the coarse graining procedure, one realizes that the system actually possesses only a finite number of degrees of freedom, recognizable in the finite number of non–vanishing singular values, which are additionally equal to one. Therefore, these embedding maps clearly satisfy the projector conditions. Combining these properties with the local interactions of the tensors implies that these fixed points describe topological theories. In fact, cylindrical consistency conditions can always be satisfied for topological theories as also discussed in chapter 7, [88]. As already mentioned this is used in the construction of the BF vacuum in (2+1)D loop quantum gravity in [175].

8.4.3 How to (possibly) get propagating degrees of freedom Despite these encouraging examples and the rich fixed point structure found in [87], we are well aware of the fact that gravity in 4D is not topological and one requires a way to arrive at a theory with propagating degrees of freedom under coarse graining. The frequently uttered route away from these topological theories is to tune the system towards a second order phase transition: at such a transition, one expects that infinitely many degrees of freedom become relevant (or can be reorganized in a non–local way). In a standard lattice approach (with a background lattice constant) one would understand this as a diverging correlation length19 . The observation that infinitely many degrees of freedom are relevant at the phase transition is in clear conflict with the truncation necessary in tensor network renormalization. To illustrate this issue and why this method necessarily has to break down at the phase transition, let us discuss the situation in quantum group spin nets. Note that these examinations are preliminary and were presented in [164], yet a much more careful analysis of the phase transitions of the systems studied in [87] is required. Despite the preliminary status of the analysis, one can make two clear observations: The first observation is that the system requires significantly more iterations for the coarse graining procedure to converge to the fixed point the closer it is fine–tuned towards the phase transition. This behaviour becomes visible if one plots a singular value over the number of iterations that can distinguish the two phases meeting at the transition: The closer the system is tuned towards the transition the longer the singular value remains (almost) constant, forming a ‘plateau’, before either flowing to one or zero depending on to which topological theory the system flows to (see also [164] for a plot). The reason why this has to happen can be drawn from the list of non–vanishing singular values, i.e. the number of relevant degrees of freedom, whose size quickly exceeds the computationally feasible bond dimension, where additionally the singular values slowly decrease in size. This immediately poses an issue to the truncation scheme, which can only provide a reasonable approximation if only irrelevant degrees of freedom are dropped. To determine whether a degree of freedom is irrelevant, the most important criterion is the size of its associated singular value compared to the most important degrees of freedom, e.g. if a singular value is several orders of magnitude smaller than the largest one. Additionally, it is advisable not to neglect the relative sizes of the smallest singular value kept and the largest truncated one. If these are very close together, one 19

The correlation function in a spin system between two spins separated by the distance r usually falls off exponentially with − rξ , where ξ is the correlation length. At a second order phase transition ξ → ∞, which implies that all spins of the system become correlated.

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8 Discussion would actually truncate a degree of freedom of similar significance as the kept one. Indeed, if both these conditions are satisfied during the whole coarse graining process, the result is consistent under increasing the accuracy of the simulations, i.e. the bond dimension of the simulation. Yet close to the phase transition, neither of these conditions can be fulfilled, in fact, no truncation can be employed without significantly altering the dynamics of the system. Hence, the necessarily employed truncation enforces a finite a number of degrees of freedom and together with local embedding maps, the system eventually flows towards a topological theory. As a side remark one can make similar observations if applying the TNR to the Ising model, which indicates that the phase transitions for the quantum group spin net models might be of second order. In condensed matter physics one is aware of this issue: Methods similar to TNR, e.g. the originally invented density matrix renormalization [116,117] method or matrix product states [118,119], were invented to (efficiently) obtain ground states for many–body quantum systems. As it turns out TNR is a well–suited approach to identify these states for ‘gapped’ systems, i.e. systems with an energy gap between the ground and the first excited state. However close to a second order phase transition, this gap disappears and the tensor network description would require an infinite bond dimension to capture the vastly spread correlation of the microscopic degrees of freedom. This behaviour is referred to as long range entanglement, which requires different methods to efficiently describe these states. One example is called multi–scale entanglement renormalization ansatz (MERA) [179] and is surprisingly analogical to the described ‘evolution from nothing’ in [88]: One degree of freedom, e.g. a qubit, is embedded via an isometry into a Hilbert space of two qubits. Then one acts with a unitary transformation, called (dis)entangler, on this two–site state, effectively entangling the two degrees of freedom. Afterwards the procedure is iteratively applied until the desired N –site state is generated, which is fully characterized by the chosen isometries and disentanglers. These maps are then specified in a variational scheme such that the expectation value of the Hamiltonian (of the system under discussion) with respect to the just constructed state is minimized. Furthermore, the locally applied entangling maps allow for an long–range entangled many–particle state20 . Therefore, methods that allow for long–range entanglement may provide a way to escape the topological fixed points encountered in [87]. In fact the importance of entangling maps has been emphasized in [88]21 as well. While tensor network renormalization does entangle the finer degrees of freedom, this only happens in a local way: In order to consecutively apply the algorithm one has to restrict the network to be regular (in terms of combinatorial information) and choose the embedding maps such that the coarse lattice has the same regular combinatorics as the fine one. Then the numerical feasibility of the algorithms dictates the introduction of a cut–off on the degrees of freedom, which in many situations is a reasonable approximation of the dynamics (and the partition function), but inevitably breaks down at a second order phase transition. Hence, non– local embedding maps, e.g. similar to MERA or the radial time–evolution sketched in chapter 7, [88], may constitute the necessary improvement to study the dynamics on the phase transition. In fact, the idea of radial time evolution in tensor networks is also related to entanglement renormalization [180], which also relies on tensor network techniques and attempts to interpret the network and the encoded entanglement as a background anti de Sitter (AdS) spacetime [181–183], related to holographic renormalization [184]. Even though this idea may appear to be fitting for gravitational theories, it may not be suitable for a dynamical theory of gravity, where one rather expects that the dynamical variables of the tensor network encode the geometric degrees of freedom, which encode a geometry in the boundary state. If one can extract this geometric information from this state, one should be able to fully reconstruct the geometry in the bulk. 20

In condensed matter, this construction scheme is usually seen from the opposite point of view: one starts with an N –site state, where the N sites are entangled on different scales. The (dis)entanglers then remove short–range entanglement between two sites, which is considered not to be universal. 21 There we particularly discuss the case of trivial refining maps, e.g. 1–4 moves in 4D Regge calculus. In that case one can show that no new physical degrees of freedom are added but gauge degrees of freedom. Thus only refining by these Pachner moves lead to configurations known as ‘stacked spheres’, which are flat and non–dynamical.

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9 Summary and conclusion In this thesis we have explored the relation between discrete approaches of quantum gravity and classical, continuous general relativity in very different situations and with very different tools, ranging from only a few basic building blocks up to literally infinitely many. While the introduced discretisation constitutes a useful tool to define and examine the dynamics of the quantum theory, it is in general not unique and obscures the relation to the classical continuum theory, e.g. by breaking diffeomorphism symmetry of general relativity [41, 42], which is deeply entwined with the dynamics of the theory. We attempt to clarify this relation by employing dynamical principles, e.g. coarse graining procedures improving the discrete theory. As we have argued in the discussion above, one can regard the results at the level of only a few simplices as guiding principles how to explore the collective dynamics of many building blocks in spin foam models. Let us summarize this line of argumentation again: At the lowest level, namely a single simplex (or vertex amplitude), we have confirmed and extended the results of previous asymptotic expansions in spin foam models [23, 25, 27–29, 99, 100] in chapters 2 and 3, see also [30] and [31]: The geometric part of the amplitude oscillates as the cosine of the discrete Regge gravity action [17, 18] associated to the simplex, constructed from representation labels of the foam. We modified the frequently used coherent state approach [29,81, 90] to also derive the complete expansion (up to leading order), in particular the missing measure factor, and derived the very first result of such a factor for the 4D Barrett–Crane spin foam model [73]. In fact, this connection between spin foams and Regge calculus at the level of one simplex is a positive direct indication that spin foam models are viable candidate quantum gravity theories. However, extensions of these results to larger triangulations are rare, see e.g. [78], even simple local changes of the underlying triangulation, so–called Pachner moves [82,83], have not been computed for 4D spin foam models (yet). Also the status of (broken) diffeomorphism symmetry in spin foam models has not been sufficiently analysed, in particular with respect to the ambiguities involved in the definition of these models. Therefore, the conjecture found in [43], stating that discrete diffeomorphism symmetry is equivalent to discretisation independence, together with the connection of spin foams and Regge calculus has motivated us to investigate triangulation independence of the (linearized) Regge path integral in chapters 4 and 5, see also [59, 84], with particular focus on the construction of a suitable path integral measure, which one might hope to be anomaly free with respect to diffeomorphisms [51]. Furthermore, the hope is to use such a measure factor as a ‘blueprint’ for spin foams. In topological 3D gravity, where the classical Regge action is triangulation independent, an almost unique path integral measure can be constructed, which is furthermore consistent with the asymptotics of the Ponzano–Regge model [23,80]. In 4D, the situation is more complicated and also more interesting: First of all, we show that even the linearized Regge action is not invariant under all Pachner moves, such that full triangulation independence cannot be achieved. Secondly, even though the calculations and Hessian matrix are strikingly similar to the 3D case, independence under the remaining subset of moves cannot be achieved either. This is due to the appearance of non–localities in the action, which do not factorise as a product of amplitudes associated to (sub)simplices. These results have several important implications: First and foremost, the (linearized) Regge action itself must necessarily be improved in order to realize a diffeomorphism symmetry in the discrete for any choice of boundary data. It has been known that the discrete diffeomorphism

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9 Summary and conclusion symmetry is only preserved if the boundary data allow for flat solutions in the bulk [44, 45], yet it surprisingly also applies to the linearized scenario. If one then attempts to improve the Regge action by a coarse graining scheme, e.g. via Pachner moves, one obtains non–local interactions already after one step, which complicates the consecutive application of the procedure. This indeed raises the question whether one can invent / apply different real space renormalization techniques [115] to spin foams that are better suited to handle non–local interactions, e.g. by using different degrees of freedom, e.g. area–angle variables [60, 142], or allowing changes on the boundary in contrast to Pachner moves, which leave the boundary unchanged. Suitable tools can be found in condensed matter theory, in particular tensor network renormalization [85, 86], which can be understood as a tool combining a collection of basic building blocks into a new effective building block; the appearance of non–localities is ‘absorbed’ into the building block itself, but is instead accompanied by an exponential growth of the boundary data under consecutive application of this method. To solve this issue in tensor network renormalization one identifies the relevant degrees of freedom of the systems (by a singular value decomposition), which allows to truncate less relevant ones and to run efficient numerical simulations. In fact, this identification of relevant degrees of freedom can be understood as a variable redefinition, dynamically relating boundary data on the fine discretisation to effective boundary data on the coarser discretisation, thus serving as an embedding map. Therefore, this tool is capable of studying the effective dynamics of the system and approximately compute the partition function and expectation values. To test the feasibility of this approach with respect to spin foam models, we have studied it on analogue spin foam models, so–called spin nets [37–39], which can be understood as dimensionally reduced spin foams, defined on the quantum group SU(2)k [122,123] in chapter 6, see also [87]. Note that these simplified models still capture the important dynamical ingredients of spin foam models, which in principle allows for more structure (by the imposition of simplicity constraints [66]) than in the related lattice gauge theories. Remarkably, we have uncovered a very rich, non–trivial fixed point structure in the spin net models, which form a regular pattern at different levels k of the quantum group, and are characterized by their excited intertwiner channels, see also [39]. Yet in contrast to [39], each of these fixed points exhibits an extended phase (in parameter space) due to the new parametrization of the initial data introduced in [87], which has been inspired by so–called intertwiner models [162]. Furthermore, the theories on the fixed points show an interesting interplay between imposing the constraints too strongly or too weakly. Yet the fixed points themselves resemble topological theories, an accompaniment of the coarse graining scheme: By enforcing a finite number of local interactions between the building blocks, the system naturally flows to a topological theory. Fortunately, close to the phase transition, one realizes that more and more degrees of freedom become relevant and the truncation scheme inevitably breaks down. Therefore we tentatively interpret this behaviour as indications that these phase transitions are of second order [164], which should lead to a theory with propagating degrees of freedom on the transition. Besides these very encouraging results for the spin foam approach, the idea of the embedding maps relating coarser and finer boundaries can be extended further (see chapter 7 and [88]): If these dynamical embedding maps satisfy so–called cylindrical consistency conditions, which are well–known in loop quantum gravity [20], one can unambiguously relate any pair of boundaries to one another. Then these tools allow us to compare states defined on different boundary Hilbert spaces by embedding them into a common refinement, e.g. the continuum, and identify them if they are identical (on this common refinement). In an inductive limit, one can thus consistently define the continuum Hilbert space of this theory. For gravitational, i.e. totally constrained, theories, this is particularly appealing, since one can then understand these embedding maps as time evolution, which do not change the physical state of the system. In this context, the physical state is merely represented on a different discretisation. Naturally, these are highly non–trivial conditions, which are very difficult to satisfy. One example, in which this is realized are topological theories, e.g. the

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previously discussed fixed points of spin net models [87]. Therefore we propose how to use these embedding maps to construct a continuum theory of quantum gravity in principle and approximately, e.g. by a coarse graining scheme like tensor network renormalization. One positive example implementing this idea has already been developed in form of the physical vacuum in (2+1)D loop quantum gravity, called the BF vacuum [175]. Of course, 3D gravity is a topological theory, but nevertheless it proves the potential of this idea. Similarly we expect this to provide the unique Hartle–Hawking vacuum [174] of the theory by starting with an ‘evolution from nothing’. Such a construction may require more non–local embedding maps that allow for long range entanglement, which might be the necessary change to escape topological theories. Again, similar tools have also been developed in condensed matter theory in the form of the ‘multi–scale entanglement renormalization ansatz’ (MERA) [179] that might prove useful in future research. To conclude this thesis, we would like to emphasize that we have made substantial progress towards defining a continuum / refinement limit of spin foam models and with it towards answering the question whether spin foam models are compatible with general relativity: we have developed new tools that help us to get a better understanding of the basic spin foam amplitudes in terms of discrete gravity, here we get a first glimpse at the chosen measure on the space of geometries. We demonstrate that in 4D discrete gravity, triangulation independence inevitably implies non–local interactions and that the Regge action and spin foam models necessarily have to be improved in order to be diffeomorphism invariant in the discrete. To effectively deal with these non–local interactions, we explore coarse graining methods developed in condensed matter theory and successfully apply them to (analogue) spin foam models, revealing a rich, interesting phase structure that indicates that spin foam models might possess intriguing continuum phases, possibly equipped with dynamics consistent with general relativity. Eventually, we extend the idea of embedding maps, which represent the key ingredients of the coarse graining procedure, to a general consistent construction scheme of the physical continuum theory from spin foam models; a ‘user guide’ how to construct quantum spacetime. Beyond doubt, there is still a long road ahead before we can confidently state that spin foam models provide a dynamics consistent with general relativity. In this thesis we provide a suggestion how this question can be investigated with positive evidence underlining the usability of the proposed tools. Following this line of thought it is necessary to extend the coarse graining methods to the full spin foam models and identify their phases. This includes computing expectation values of observables, correlation functions etc. necessary to characterize the systems. We also feel obliged to mention that the systems under discussion in the entirety of this thesis describe pure gravity and in order to check consistency with general relativity also includes coupling matter to the quantum gravity theory. As a closing remark, let us emphasize again the importance of checking the consistency with the classical theory: Due to the lack of experimental data and therefore a lack of guidance concerning what a theory of quantum gravity must capture, e.g. what are the physical degrees of freedom to quantize, several different approaches to quantum gravity have been developed, each of them with their own premises, advantages and disadvantages. A priori the question which of these research programs is to be favoured cannot be answered and relies heavily on personal opinion. Therefore it is indispensable to make an effort to derive predictions, check consistency with the classical theory, in short, establish a connection to physical reality, such that the theory can be verified or falsified. To accomplish such a task is profoundly ambitious, yet progress might be in reach if one broadens one’s horizon and adopts ideas developed in different fields. To be more concrete, we have been able to investigate spin foam models in a novel way, uncovering new insights by using condensed matter techniques; future developments may prove to be mutually beneficial. Therefore, one should not be afraid to reach out for new methods and testable predictions, since even if the outcome is negative, the knowledge that an approach does not work and why is a substantial contribution to

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9 Summary and conclusion science that should not be underestimated.

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